Carath\'eodory number of homogeneous convex cones

TL;DR

This paper investigates the Carathéodory number of homogeneous convex cones using spectrahedral representations, providing characterizations and conditions for self-duality and sparsity related to chordal graphs.

Contribution

It offers a new characterization of homogeneous convex cones where rank equals Carathéodory number and links self-duality to this property.

Findings

Homogeneous convex cones have ranks matching their Carathéodory numbers under certain conditions.

Self-duality of a cone is characterized by the equality of ranks and Carathéodory numbers of the cone and its dual.

Sparse spectrahedral cones that are homogeneous are precisely those associated with homogeneous chordal graphs.

Abstract

We study the Carath\'eodory number of homogeneous convex cones via their spectrahedral representations. A characterization of homogeneous convex cones whose ranks match their Carath\'eodory numbers is given. This characterization is then used to show that a homogeneous convex cone is selfdual if and only if its rank matches the Carath\'eodory numbers of both its closure and its dual cone. It is further used to show that the only sparse spectrahedral cones that are homogeneous convex cones are those described by homogeneous chordal graphs.