Question 11106: Coordinatewise Universal Convexity of a Closed Box

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. 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?).

• 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. 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?).

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

box coordinate0:closed bounds=[-7,8];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--7>=0;y--7>=0;8-x>=0;8-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-7,8]

z--7=lambda*(x--7)+(1-lambda)*(y--7)>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-7,8]

8-z=lambda*(8-x)+(1-lambda)*(8-y)>=0

box coordinate0:both symbolic slack identities hold for every chosen x,y,lambda

all x,y in [-7,8] and all lambda in [0,1] give z in [-7,8]

box coordinate1:closed bounds=[-6,9];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--6>=0;y--6>=0;9-x>=0;9-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-6,9]

z--6=lambda*(x--6)+(1-lambda)*(y--6)>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-6,9]

9-z=lambda*(9-x)+(1-lambda)*(9-y)>=0

box coordinate1:both symbolic slack identities hold for every chosen x,y,lambda

all x,y in [-6,9] and all lambda in [0,1] give z in [-6,9]

box bounds=[-7, 8, -6, 9];both coordinate interval certificates hold with the same arbitrary lambda

both coordinates remain in their intervals;whole closed box convex=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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

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