# Question 11120: Coordinatewise Universal Convexity of a Closed Box [Return to catalogue page 56](https://ocme.evemisslab.com/en/questions/page/56/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r07-convex-set-foundations/#q-20261013-advance-20261024-r07-convex-set-0045) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0021-11120 · q-20261013-advance-20261024-r07-convex-set-0045 ## 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](https://ocme.evemisslab.com/en/mko/mko-convex-set/) ## Step-by-step derivation S001 · impliesBefore box coordinate0:closed bounds=[-21,22];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1 After lambda>=0;1-lambda>=0;x--21>=0;y--21>=0;22-x>=0;22-y>=0 Conditions: none Choose arbitrary interval members and any permitted lambda.All four endpoint slacks are nonnegative;equal bounds are valid. Step identifiers and dependenciesRule: universal_interval_arbitrary_members · Uses: G001 S002 · impliesBefore z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-21,22] After z--21=lambda*(x--21)+(1-lambda)*(y--21)>=0 Conditions: none Use the weights'unit sum to expand z minus the lower bound as a sum of nonnegative products.This is not sampling. Step identifiers and dependenciesRule: universal_interval_lower_identity · Uses: G001, S001 S003 · impliesBefore z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-21,22] After 22-z=lambda*(22-x)+(1-lambda)*(22-y)>=0 Conditions: none Likewise the upper bound minus z is nonnegative,so both bounding inequalities hold. Step identifiers and dependenciesRule: universal_interval_upper_identity · Uses: G001, S002 S004 · impliesBefore box coordinate0:both symbolic slack identities hold for every chosen x,y,lambda After all x,y in [-21,22] and all lambda in [0,1] give z in [-21,22] Conditions: none No special restriction was added to the arbitrary endpoints or lambda.Discharge their universal quantifiers to obtain convexity of the whole closed interval. Step identifiers and dependenciesRule: universal_interval_quantifier_discharge · Uses: G001, S003 S005 · impliesBefore box coordinate1:closed bounds=[-20,23];arbitrary x,y in these bounds;arbitrary 0<=lambda<=1 After lambda>=0;1-lambda>=0;x--20>=0;y--20>=0;23-x>=0;23-y>=0 Conditions: none Choose arbitrary interval members and any permitted lambda.All four endpoint slacks are nonnegative;equal bounds are valid. Step identifiers and dependenciesRule: universal_interval_arbitrary_members · Uses: G001, S004 S006 · impliesBefore z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,23] After z--20=lambda*(x--20)+(1-lambda)*(y--20)>=0 Conditions: none Use the weights'unit sum to expand z minus the lower bound as a sum of nonnegative products.This is not sampling. Step identifiers and dependenciesRule: universal_interval_lower_identity · Uses: G001, S005 S007 · impliesBefore z=lambda*x+(1-lambda)*y;lambda+(1-lambda)=1;bounds=[-20,23] After 23-z=lambda*(23-x)+(1-lambda)*(23-y)>=0 Conditions: none Likewise the upper bound minus z is nonnegative,so both bounding inequalities hold. Step identifiers and dependenciesRule: universal_interval_upper_identity · Uses: G001, S006 S008 · impliesBefore box coordinate1:both symbolic slack identities hold for every chosen x,y,lambda After all x,y in [-20,23] and all lambda in [0,1] give z in [-20,23] Conditions: none No special restriction was added to the arbitrary endpoints or lambda.Discharge their universal quantifiers to obtain convexity of the whole closed interval. Step identifiers and dependenciesRule: universal_interval_quantifier_discharge · Uses: G001, S007 S009 · impliesBefore box bounds=[-21, 22, -20, 23];both coordinate interval certificates hold with the same arbitrary lambda After both coordinates remain in their intervals;whole closed box convex=Yes Conditions: none Apply each interval certificate to the same arbitrary lambda.Both coordinates remain within bounds,hence in the box.Zero widths are included. Step identifiers and dependenciesRule: box_coordinatewise_universal_conjunction · Uses: G001, S008 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: dc992ddc9f264ac39d61d4c1c808572a7f90c3183366efe4c7c97595bdd705b3 - candidate record SHA-256: 8b9f181c49fd258cd8f2860c77b83fb8e70234822534cf2aa379883ef5bdb782 - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r07-convex-set-0045.json)