A "hidden" characterization of polyhedral convex sets

TL;DR

This paper characterizes polyhedral convex sets in complete linear metric spaces by a geometric property involving the inability of infinite subsets outside the set to be hidden behind it, providing a new perspective on convex geometry.

Contribution

It introduces a novel characterization of polyhedral convex sets based on the concept of hidden subsets, linking geometric properties with convex set structure.

Findings

Characterization of polyhedral convex sets via hidden subsets

Equivalence between polyhedrality and the absence of hidden infinite subsets

Provides a new geometric criterion for polyhedral convexity

Abstract

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not = \emptyset$ for any distinct points $x,y\in A$.