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Universal Interval, Box and Halfspace Certificates versus Midpoint Counterexamples

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261024-r07-convex-set-foundations · 2026-10-13 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

1. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-2, upper=3, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-2, 3, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-2, 3, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

2. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-3, upper=4, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-3, 4, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-3, 4, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

3. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-4, upper=5, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-4, 5, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-4, 5, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

4. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-5, upper=6, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-5, 6, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-5, 6, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

5. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-6, upper=7, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-6, 7, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-6, 7, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

6. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-7, upper=8, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-7, 8, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-7, 8, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

7. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-8, upper=9, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-8, 9, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-8, 9, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

8. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-9, upper=10, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-9, 10, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-9, 10, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

9. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-10, upper=11, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-10, 11, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-10, 11, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

10. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-11, upper=12, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-11, 12, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-11, 12, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

11. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-12, upper=13, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-12, 13, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-12, 13, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

12. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-13, upper=14, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-13, 14, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-13, 14, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

13. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-14, upper=15, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-14, 15, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-14, 15, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

14. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-15, upper=16, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-15, 16, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-15, 16, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

15. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-16, upper=17, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-16, 17, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-16, 17, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

16. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-17, upper=18, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-17, 18, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-17, 18, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

17. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-18, upper=19, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-18, 19, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-18, 19, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

18. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-19, upper=20, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-19, 20, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-19, 20, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

19. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-20, upper=21, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-20, 21, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-20, 21, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

20. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-21, upper=22, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-21, 22, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-21, 22, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

21. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-22, upper=23, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-22, 23, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-22, 23, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

22. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-23, upper=24, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-23, 24, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-23, 24, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

23. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-24, upper=25, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-24, 25, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-24, 25, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

24. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-25, upper=26, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-25, 26, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-25, 26, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

25. A Universal Closed-Interval Convexity Certificate

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Nonempty closed interval [lower,upper] has lower=-26, upper=27, allowing equality. Give (lower,upper,all x,y in interval and all lambda in [0,1],convex?).

View answer and explanation

(-26, 27, all x,y in interval and all lambda in [0,1], Yes)

For arbitrary x,y in the interval and 0≤lambda≤1, multiplying both endpoint inequalities by nonnegative lambda and 1−lambda and adding keeps their weighted sum between the bounds. This covers every endpoint pair and lambda, not samples. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (-26, 27, all x,y in interval and all lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-interval-certificate · candidate

26. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-2, 3, -1, 4] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 3, -1, 4], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

27. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-3, 4, -2, 5] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 4, -2, 5], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

28. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-4, 5, -3, 6] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 5, -3, 6], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

29. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-5, 6, -4, 7] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 6, -4, 7], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

30. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-6, 7, -5, 8] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 7, -5, 8], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

31. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-7, 8, -6, 9] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 8, -6, 9], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

32. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-8, 9, -7, 10] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 9, -7, 10], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

33. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-9, 10, -8, 11] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 10, -8, 11], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

34. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-10, 11, -9, 12] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 11, -9, 12], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

35. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-11, 12, -10, 13] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 12, -10, 13], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

36. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-12, 13, -11, 14] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 13, -11, 14], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

37. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-13, 14, -12, 15] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 14, -12, 15], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

38. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-14, 15, -13, 16] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 15, -13, 16], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

39. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-15, 16, -14, 17] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 16, -14, 17], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

40. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-16, 17, -15, 18] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 17, -15, 18], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

41. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-17, 18, -16, 19] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 18, -16, 19], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

42. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-18, 19, -17, 20] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 19, -17, 20], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

43. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-19, 20, -18, 21] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 20, -18, 21], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

44. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-20, 21, -19, 22] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 21, -19, 22], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

45. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-21, 22, -20, 23] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 22, -20, 23], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

46. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-22, 23, -21, 24] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23, -21, 24], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

47. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-23, 24, -22, 25] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 24, -22, 25], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

48. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-24, 25, -23, 26] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 25, -23, 26], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

49. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-25, 26, -24, 27] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 26, -24, 27], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

50. Coordinatewise Universal Convexity of a Closed Box

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Bounds [x_min,x_max,y_min,y_max]=[-26, 27, -25, 28] are ordered and may have zero widths. Give (bounds,coordinatewise interval certificate for every lambda in [0,1],convex?).

View answer and explanation

([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], Yes)

Apply the universal interval inequality independently in each coordinate. Their conjunction covers every point pair and every permitted lambda in the box. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 27, -25, 28], coordinatewise interval certificate for every lambda in [0,1], Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-box-certificate · candidate

51. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=2, left=[0, 0], right=[1, 0], lambda=1/3∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

52. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=3, left=[0, 0], right=[2, 0], lambda=2/4∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 3, [0, 0], [2, 0], 0, 2, 1/2, [1, 0], 1, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

53. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=4, left=[0, 0], right=[3, 0], lambda=3/5∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 4, [0, 0], [3, 0], 0, 3, 3/5, [6/5, 0], 6/5, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

54. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=5, left=[0, 0], right=[4, 0], lambda=4/6∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 5, [0, 0], [4, 0], 0, 4, 2/3, [4/3, 0], 4/3, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

55. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=6, left=[0, 0], right=[5, 0], lambda=5/7∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 6, [0, 0], [5, 0], 0, 5, 5/7, [10/7, 0], 10/7, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

56. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=7, left=[0, 0], right=[6, 0], lambda=6/8∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 7, [0, 0], [6, 0], 0, 6, 3/4, [3/2, 0], 3/2, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

57. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=8, left=[0, 0], right=[7, 0], lambda=7/9∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 8, [0, 0], [7, 0], 0, 7, 7/9, [14/9, 0], 14/9, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

58. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=9, left=[0, 0], right=[8, 0], lambda=8/10∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 9, [0, 0], [8, 0], 0, 8, 4/5, [8/5, 0], 8/5, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

59. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=10, left=[0, 0], right=[9, 0], lambda=9/11∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 10, [0, 0], [9, 0], 0, 9, 9/11, [18/11, 0], 18/11, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

60. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=11, left=[0, 0], right=[10, 0], lambda=10/12∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 11, [0, 0], [10, 0], 0, 10, 5/6, [5/3, 0], 5/3, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

61. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=12, left=[0, 0], right=[11, 0], lambda=11/13∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 12, [0, 0], [11, 0], 0, 11, 11/13, [22/13, 0], 22/13, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

62. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=13, left=[0, 0], right=[12, 0], lambda=12/14∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 13, [0, 0], [12, 0], 0, 12, 6/7, [12/7, 0], 12/7, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

63. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=14, left=[0, 0], right=[13, 0], lambda=13/15∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 14, [0, 0], [13, 0], 0, 13, 13/15, [26/15, 0], 26/15, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

64. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=15, left=[0, 0], right=[14, 0], lambda=14/16∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 15, [0, 0], [14, 0], 0, 14, 7/8, [7/4, 0], 7/4, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

65. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=16, left=[0, 0], right=[15, 0], lambda=15/17∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 16, [0, 0], [15, 0], 0, 15, 15/17, [30/17, 0], 30/17, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

66. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=17, left=[0, 0], right=[16, 0], lambda=16/18∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 17, [0, 0], [16, 0], 0, 16, 8/9, [16/9, 0], 16/9, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

67. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=18, left=[0, 0], right=[17, 0], lambda=17/19∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 18, [0, 0], [17, 0], 0, 17, 17/19, [34/19, 0], 34/19, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

68. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=19, left=[0, 0], right=[18, 0], lambda=18/20∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 19, [0, 0], [18, 0], 0, 18, 9/10, [9/5, 0], 9/5, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

69. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=20, left=[0, 0], right=[19, 0], lambda=19/21∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 20, [0, 0], [19, 0], 0, 19, 19/21, [38/21, 0], 38/21, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

70. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=21, left=[0, 0], right=[20, 0], lambda=20/22∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 21, [0, 0], [20, 0], 0, 20, 10/11, [20/11, 0], 20/11, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

71. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=22, left=[0, 0], right=[21, 0], lambda=21/23∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 22, [0, 0], [21, 0], 0, 21, 21/23, [42/23, 0], 42/23, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

72. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=23, left=[0, 0], right=[22, 0], lambda=22/24∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 23, [0, 0], [22, 0], 0, 22, 11/12, [11/6, 0], 11/6, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

73. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=24, left=[0, 0], right=[23, 0], lambda=23/25∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 24, [0, 0], [23, 0], 0, 23, 23/25, [46/25, 0], 46/25, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

74. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=25, left=[0, 0], right=[24, 0], lambda=24/26∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 25, [0, 0], [24, 0], 0, 24, 12/13, [24/13, 0], 24/13, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

75. Linear Halfspace Closure at Qualified Endpoints

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=26, left=[0, 0], right=[25, 0], lambda=25/27∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, Yes)

Linearity makes the weighted-point dot product the same nonnegative weighted sum of endpoint dot products, hence at most the bound. Boundary equality is included. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-halfspace-combination · candidate

76. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [2, 0], [0, 2]] retain input order; endpoints left=[2, 0], right=[0, 2] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [2, 0], [0, 2]], [2, 0], [0, 2], [1, 1], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

77. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [4, 0], [0, 4]] retain input order; endpoints left=[4, 0], right=[0, 4] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [4, 0], [0, 4]], [4, 0], [0, 4], [2, 2], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

78. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [6, 0], [0, 6]] retain input order; endpoints left=[6, 0], right=[0, 6] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [6, 0], [0, 6]], [6, 0], [0, 6], [3, 3], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

79. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [8, 0], [0, 8]] retain input order; endpoints left=[8, 0], right=[0, 8] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [8, 0], [0, 8]], [8, 0], [0, 8], [4, 4], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

80. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [10, 0], [0, 10]] retain input order; endpoints left=[10, 0], right=[0, 10] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [10, 0], [0, 10]], [10, 0], [0, 10], [5, 5], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

81. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [12, 0], [0, 12]] retain input order; endpoints left=[12, 0], right=[0, 12] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [12, 0], [0, 12]], [12, 0], [0, 12], [6, 6], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

82. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [14, 0], [0, 14]] retain input order; endpoints left=[14, 0], right=[0, 14] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [14, 0], [0, 14]], [14, 0], [0, 14], [7, 7], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

83. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [16, 0], [0, 16]] retain input order; endpoints left=[16, 0], right=[0, 16] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [16, 0], [0, 16]], [16, 0], [0, 16], [8, 8], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

84. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [18, 0], [0, 18]] retain input order; endpoints left=[18, 0], right=[0, 18] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [18, 0], [0, 18]], [18, 0], [0, 18], [9, 9], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

85. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [20, 0], [0, 20]] retain input order; endpoints left=[20, 0], right=[0, 20] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [20, 0], [0, 20]], [20, 0], [0, 20], [10, 10], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

86. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [22, 0], [0, 22]] retain input order; endpoints left=[22, 0], right=[0, 22] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [22, 0], [0, 22]], [22, 0], [0, 22], [11, 11], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

87. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [24, 0], [0, 24]] retain input order; endpoints left=[24, 0], right=[0, 24] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [24, 0], [0, 24]], [24, 0], [0, 24], [12, 12], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

88. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [26, 0], [0, 26]] retain input order; endpoints left=[26, 0], right=[0, 26] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [26, 0], [0, 26]], [26, 0], [0, 26], [13, 13], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

89. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [28, 0], [0, 28]] retain input order; endpoints left=[28, 0], right=[0, 28] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [28, 0], [0, 28]], [28, 0], [0, 28], [14, 14], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

90. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [30, 0], [0, 30]] retain input order; endpoints left=[30, 0], right=[0, 30] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [30, 0], [0, 30]], [30, 0], [0, 30], [15, 15], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

91. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [32, 0], [0, 32]] retain input order; endpoints left=[32, 0], right=[0, 32] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [32, 0], [0, 32]], [32, 0], [0, 32], [16, 16], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

92. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [34, 0], [0, 34]] retain input order; endpoints left=[34, 0], right=[0, 34] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [34, 0], [0, 34]], [34, 0], [0, 34], [17, 17], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

93. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [36, 0], [0, 36]] retain input order; endpoints left=[36, 0], right=[0, 36] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [36, 0], [0, 36]], [36, 0], [0, 36], [18, 18], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

94. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [38, 0], [0, 38]] retain input order; endpoints left=[38, 0], right=[0, 38] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [38, 0], [0, 38]], [38, 0], [0, 38], [19, 19], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

95. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [40, 0], [0, 40]] retain input order; endpoints left=[40, 0], right=[0, 40] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [40, 0], [0, 40]], [40, 0], [0, 40], [20, 20], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

96. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [42, 0], [0, 42]] retain input order; endpoints left=[42, 0], right=[0, 42] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [42, 0], [0, 42]], [42, 0], [0, 42], [21, 21], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

97. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [44, 0], [0, 44]] retain input order; endpoints left=[44, 0], right=[0, 44] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [44, 0], [0, 44]], [44, 0], [0, 44], [22, 22], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

98. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [46, 0], [0, 46]] retain input order; endpoints left=[46, 0], right=[0, 46] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [46, 0], [0, 46]], [46, 0], [0, 46], [23, 23], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

99. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [48, 0], [0, 48]] retain input order; endpoints left=[48, 0], right=[0, 48] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [48, 0], [0, 48]], [48, 0], [0, 48], [24, 24], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

100. A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. Distinct finite points=[[0, 0], [50, 0], [0, 50]] retain input order; endpoints left=[50, 0], right=[0, 50] must belong to the set. Give (points,left,right,exact midpoint,midpoint present?,is this a nonconvexity witness?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs.

View answer and explanation

([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], No, Yes)

An absent midpoint between two members is one exact counterexample. A present midpoint for this pair does not prove the whole set convex; another pair or weight may still fail. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is ([[0, 0], [50, 0], [0, 50]], [50, 0], [0, 50], [25, 25], No, Yes).

Reviewed step-by-step derivation

Convex set · qf-mod24-convexset-finite-midpoint-counterexample · candidate

Canonical question JSONL · English JSONL · Validation report