Face relative interior of convex sets in topological vector spaces

TL;DR

This paper introduces a new concept called face relative interior for convex sets in topological vector spaces, which better captures their geometry than existing notions, and establishes its properties and relations.

Contribution

It proposes the face relative interior, a novel geometric concept, and analyzes its properties, nonemptiness conditions, and relations to other convex set interiors.

Findings

Face relative interior partitions convex sets into face relative interiors of their closure-equivalent faces.

Conditions for nonemptiness of face relative interior are established.

Basic calculus rules for face relative interior are proved.

Abstract

A new notion of face relative interior for convex sets in topological real vector spaces is introduced in this work. Face relative interior is grounded in the facial structure, and may capture the geometry of convex sets in topological vector spaces better than other generalisations of relative interior. We show that the face relative interior partitions convex sets into face relative interiors of their closure-equivalent faces (different to the partition generated by intrinsic cores), establish the conditions for nonemptiness of this new notion, compare the face relative interior with other concepts of convex interior and prove basic calculus rules.