Inner approximations of convex sets and intersections of projectionally exposed cones

TL;DR

This paper explores the properties of projectionally exposed convex cones, demonstrating that in five dimensions, the intersection of such cones may not be p-exposed, and introduces a new technique for inner approximations with specific facial structures.

Contribution

It constructs the first examples of p-exposed cones in dimension five whose intersection is not p-exposed and develops a novel method for inner convex approximations with controlled facial features.

Findings

In dimension five, the intersection of two p-exposed cones may not be p-exposed.

Introduces a new technique for constructing inner convex approximations with specific facial structures.

Shows that properties coinciding in dimension four differ in dimension five.

Abstract

A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper we construct two p-exposed cones in dimension $5$ whose intersection is not p-exposed. This construction also leads to the first example of an amenable cone that is not projectionally exposed, showing that these properties, which coincide in dimension at most $4$, are distinct in dimension $5$. In order to achieve these goals, we develop a new technique for constructing arbitrarily tight inner convex approximations of compact convex sets with desired facial structure. These inner approximations have the property that all proper faces are extreme points, with the exception of a specific exposed face of the original set.