Amenable cones are particularly nice

TL;DR

This paper explores the properties of amenable convex cones, establishing their relationships with other cone properties, and provides examples and open questions in the geometric analysis of convex cones.

Contribution

It establishes new properties of amenable cones, their relationships with niceness and projectional exposure, and constructs examples distinguishing these properties.

Findings

Amenability of a cone is equivalent to that of its compact slices.

Homogeneous and doubly nonnegative cones are amenable.

An example of a nice but not amenable cone is constructed.

Abstract

Amenability is a geometric property of convex cones that is stronger than facial exposedness and assists in the study of error bounds for conic feasibility problems. In this paper we establish numerous properties of amenable cones, and investigate the relationships between amenability and other properties of convex cones, such as niceness and projectional exposure. We show that the amenability of a compact slice of a closed convex cone is equivalent to the amenability of the cone, and prove several results on the preservation of amenability under intersections and other convex operations. It then follows that homogeneous, doubly nonnegative and other cones that can be represented as slices of the cone of positive semidefinite matrices, are amenable. It is known that projectionally exposed cones are amenable and that amenable cones are nice, however the converse statements have been open questions. We construct an example of a four-dimensional cone that is nice but not amenable. We also show that amenable cones are projectionally exposed in dimensions up to and including four. We conclude with a discussion on open problems related to facial structure of convex sets that we came across in the course of this work, but were not able to fully resolve.