# Convex set

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set.

mko-convex-set · definition · v0.10.0



## Mathematical statement

A subset C of a real vector space is convex if lambda*x+(1−lambda)*y belongs to C for every x,y in C and every lambda in [0,1]. Positive cases here use complete algebraic certificates for intervals, boxes and halfspaces; a single exact counterexample can refute convexity.





## Representative formula (exact TeX)

K(C)





## Assumptions


- Only nonempty closed intervals, nonempty axis-aligned boxes, closed halfspaces and explicit finite-point-set counterexamples are handled.
- Sampling finitely many lambda values cannot prove convexity. Failure of a proposed finite-point counterexample does not conversely establish convexity.





## Symbol scope


- K — convexity certificate or counterexample
- C — real-vector-space subset





## Intuition, definition, and misconceptions

Coordinatewise upper- and lower-bound certificates cover every lambda for intervals and boxes.

A linear inequality defining a halfspace is preserved by convex combinations.

If a set contains both endpoints but not their midpoint, that single midpoint witness suffices to refute convexity.

Do not assert convexity after a few lambda checks or merely because one counterexample search failed.





## Adopted derivation guide

[Convex set: complete convexity certificates for a halfspace and an axis-aligned box, and a midpoint counterexample for a finite point set](https://ocme.evemisslab.com/en/derivations/derivation-mko-convex-set-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite convex combination](https://ocme.evemisslab.com/en/mko/mko-finite-convex-combination/) — Two-point convex combinations describe segments.
- [Subset](https://ocme.evemisslab.com/en/mko/mko-subset/) — A convex set is a subset of a real vector space.
- [Real vector space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) — Segment operations use vector addition and scalar multiplication.





## Verification boundary

Positive cases require universal algebraic certificates; a failed counterexample is not a positive proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add convex functions, convex hulls and convex optimization.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/convex-set.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-convex-set/index.md)