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CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31verified 2026-08-08 (gpt-5.6-terra-codex-subscription)
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Every nonempty convex subset of Rn\mathbb{R}^n is contractible

Statement

Let n≥1n\ge1. Every nonempty convex subset C⊆RnC\subseteq\mathbb R^n, with its Euclidean subspace topology, is contractible. More precisely, for each c∈Cc\in C the formula

H(x,t)=(1−t)x+tcH(x,t)=(1-t)x+tc

is a homotopy from id⁡C\operatorname{id}_C to the constant map at cc.

Facts & Assumptions

Given: A nonempty convex subset C⊆RnC\subseteq\mathbb R^n and a point c∈Cc\in C.

[L1]

Any two continuous maps into a nonempty convex subset of Rn\mathbb R^n are homotopic by the straight-line formula (Any two continuous maps into a nonempty convex subset of Rn\mathbb{R}^n are homotopic by straight lines).

[L2]

A nonempty space is contractible exactly when its identity map is nullhomotopic (A nonempty space is contractible if and only if its identity map is nullhomotopic).

Proof

technique · direct
1.1

Apply [L1] to id⁡C:C→C\operatorname{id}_C:C\to C and the constant map cc:C→Cc_c:C\to C. The resulting homotopy is H(x,t)=(1−t)x+tcH(x,t)=(1-t)x+tc.

L1
2.1

Thus id⁡C\operatorname{id}_C is nullhomotopic, so CC is contractible by [L2].

step 1.1L2∎

Depends on

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