Question 11141: A Missing Midpoint Refutes Convexity, but a Present One Does Not Prove It

Original question and givens

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.

• G001 — 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.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-convex-set

Step-by-step derivation

ordered finite point set=[[0, 0], [32, 0], [0, 32]];left=[32, 0];right=[0, 32]

both endpoints belong;choose lambda=1/2

midpoint coordinate0 numerator := sum(32, 0)

midpoint coordinate0 numerator = 32

midpoint coordinate0 := div(32, 2)

midpoint coordinate0 = 16

midpoint coordinate1 numerator := sum(0, 32)

midpoint coordinate1 numerator = 32

midpoint coordinate1 := div(32, 2)

midpoint coordinate1 = 16

candidate point index=0;point=[0, 0];midpoint=[16, 16]

all coordinates equal=No

candidate point index=1;point=[32, 0];midpoint=[16, 16]

all coordinates equal=No

candidate point index=2;point=[0, 32];midpoint=[16, 16]

all coordinates equal=No

all point comparisons=[No, No, No]

midpoint present=No;this nonconvex witness=Yes

Reverse checks

• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute midpoint coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute midpoint coordinate1 numerator with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute midpoint coordinate0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute midpoint coordinate0 numerator with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

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

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.