Question 11120: 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]=[-21, 22, -20, 23] 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]=[-21, 22, -20, 23] 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=[-21,22];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--21>=0;y--21>=0;22-x>=0;22-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-21,22]

z--21=lambda*(x--21)+(1-lambda)*(y--21)>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-21,22]

22-z=lambda*(22-x)+(1-lambda)*(22-y)>=0

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

all x,y in [-21,22] and all lambda in [0,1] give z in [-21,22]

box coordinate1:closed bounds=[-20,23];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--20>=0;y--20>=0;23-x>=0;23-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,23]

z--20=lambda*(x--20)+(1-lambda)*(y--20)>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,23]

23-z=lambda*(23-x)+(1-lambda)*(23-y)>=0

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

all x,y in [-20,23] and all lambda in [0,1] give z in [-20,23]

box bounds=[-21, 22, -20, 23];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

([-21, 22, -20, 23], 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.