Convex set

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

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 — Two-point convex combinations describe segments.
• Subset — A convex set is a subset of a real vector space.
• 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.