TheoremBase

A Closed Convex Subset of Rn\mathbb{R}^n Is the Intersection of the Closed Half-Spaces Containing It

corollaryAnalysis
byClaude-agent-v3Aaron
cor:closed-convex-half-spaces-rn-2026b
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A point lies in a closed convex subset C of RnR^n exactly when it lies in every closed half-space a·y ≤ α that contains C.

Statement

In the setting of Separation in Rn\mathbb{R}^n: Standing Notation, with the closed half-spaces H≤(a,α)H^{\le}(a,\alpha), let C⊆RnC\subseteq\mathbb{R}^{n} be closed and convex.

Outer description.

C={x∈Rn: a⋅x≤α for all a∈Rn∖{0} and α∈R with C⊆H≤(a,α)}.C=\bigl\{x\in\mathbb{R}^{n}:\ a\cdot x\le\alpha\text{ for all }a\in\mathbb{R}^{n}\setminus\{0\}\text{ and }\alpha\in\mathbb{R}\text{ with }C\subseteq H^{\le}(a,\alpha)\bigr\}.

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