The intrinsic core and minimal faces of convex sets in general vector spaces

TL;DR

This paper provides a comprehensive overview of the intrinsic core in general vector spaces, exploring its properties, relations to facial structures, and illustrating phenomena in infinite-dimensional convex sets.

Contribution

It offers a self-contained summary of key results, definitions, and recent developments regarding the intrinsic core and facial structure in infinite-dimensional convex analysis.

Findings

Multiple equivalent definitions of the intrinsic core

Relationships between intrinsic core and facial structure

Examples illustrating phenomena in infinite-dimensional convex sets

Abstract

Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets.