Question 11119: 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]=[-20, 21, -19, 22] 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]=[-20, 21, -19, 22] 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=[-20,21];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--20>=0;y--20>=0;21-x>=0;21-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,21]

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

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,21]

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

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

all x,y in [-20,21] and all lambda in [0,1] give z in [-20,21]

box coordinate1:closed bounds=[-19,22];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1

lambda>=0;1-lambda>=0;x--19>=0;y--19>=0;22-x>=0;22-y>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-19,22]

z--19=lambda*(x--19)+(1-lambda)*(y--19)>=0

z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-19,22]

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

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

all x,y in [-19,22] and all lambda in [0,1] give z in [-19,22]

box bounds=[-20, 21, -19, 22];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

([-20, 21, -19, 22], 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.