Question 11151: Linear Halfspace Closure at Qualified Endpoints

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. 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.

• 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. 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.

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

left endpoint dot term0 := mul(1, 0)

left endpoint dot term0 = 0

left endpoint dot term1 := mul(1, 0)

left endpoint dot term1 = 0

left endpoint dot sum := sum(0, 0)

left endpoint dot sum = 0

right endpoint dot term0 := mul(1, 1)

right endpoint dot term0 = 1

right endpoint dot term1 := mul(1, 0)

right endpoint dot term1 = 0

right endpoint dot sum := sum(1, 0)

right endpoint dot sum = 1

normal=[1, 1];bound=2;left dot=0;right dot=1

both endpoints satisfy the closed halfspace inequality

halfspace point complementary weight := sub(1, 1/3)

halfspace point complementary weight = 2/3

halfspace point:raw lambda=[1, 3];lambda=1/3;one-minus-lambda=2/3

in_closed_unit_interval=Yes;lambda weights left

halfspace point:ordered vectors=[[0, 0], [1, 0]];aligned weights=[1/3, 2/3]

dimension=2;preserve duplicates and order

halfspace point coordinate0 term0 := mul(1/3, 0)

halfspace point coordinate0 term0 = 0

halfspace point coordinate0 term1 := mul(2/3, 1)

halfspace point coordinate0 term1 = 2/3

halfspace point coordinate0 sum := sum(0, 2/3)

halfspace point coordinate0 sum = 2/3

halfspace point coordinate1 term0 := mul(1/3, 0)

halfspace point coordinate1 term0 = 0

halfspace point coordinate1 term1 := mul(2/3, 0)

halfspace point coordinate1 term1 = 0

halfspace point coordinate1 sum := sum(0, 0)

halfspace point coordinate1 sum = 0

combined point dot term0 := mul(1, 2/3)

combined point dot term0 = 2/3

combined point dot term1 := mul(1, 0)

combined point dot term1 = 0

combined point dot sum := sum(2/3, 0)

combined point dot sum = 2/3

halfspace bound slack := sub(2, 2/3)

halfspace bound slack = 4/3

normal dot z=2/3;endpoint dots=(0,1);lambda=1/3;bound=2

normal dot z=lambda*0+(1-lambda)*1<=2;inside=Yes

Reverse checks

• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute halfspace bound slack with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute combined point dot sum with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute combined point dot term1 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute combined point dot term0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute halfspace point coordinate1 sum with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute halfspace point coordinate1 term1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute halfspace point coordinate1 term0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute halfspace point coordinate0 sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute halfspace point coordinate0 term1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute halfspace point coordinate0 term0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Recompute halfspace point complementary weight with exact rational arithmetic, not rounded decimals.
• 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 — Recompute right endpoint dot sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute right endpoint dot term1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute right endpoint dot term0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute left endpoint dot sum with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute left endpoint dot term1 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute left endpoint dot term0 with exact rational arithmetic, not rounded decimals.

Result

([1, 1], 2, [0, 0], [1, 0], 0, 1, 1/3, [2/3, 0], 2/3, 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.