Pointed Closed Convex Sets are the Intersection of All Rational Supporting Closed Halfspaces

TL;DR

This paper proves that any pointed closed convex set in Euclidean space can be represented as the intersection of all rational closed halfspaces containing it, extending previous results from compact convex sets.

Contribution

The authors generalize a known representation of convex sets by rational halfspaces from compact to pointed closed convex sets in Euclidean space.

Findings

Any pointed closed convex set equals the intersection of all rational supporting halfspaces.

The result extends previous work limited to compact convex sets.

Provides a new characterization of convex sets via rational halfspaces.

Abstract

We prove that every pointed closed convex set in $\mathbb{R}^n$ is the intersection of all the rational closed halfspaces that contain it. This generalizes a previous result by the authors for compact convex sets.