A bound on the Carath\'eodory number

TL;DR

This paper refines the upper bound on the Carathéodory number of convex cones by relating it to the length of the longest chain of faces, providing tight bounds for certain cone families and connecting to matrix factorization concepts.

Contribution

It establishes a sharper bound on the Carathéodory number based on face chains, proves its tightness for specific cones, and links it to symmetric cones and cp-rank of matrices.

Findings

Bound k(K) <= l-1, where l is the face chain length.

The bound is tight for symmetric and smooth cones.

Counterexamples show the bound can be non-sharp.

Abstract

The Carath\'eodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carath\'eodory's Theorem gives the bound k(K) <= dim (K). In this work we observe that this bound can be sharpened to k(K) <= l-1, where l is the length of the longest chain of nonempty faces contained in K, thus tying the Carath\'eodory number with a key quantity that appears in the analysis of facial reduction algorithms. We show that this bound is tight for several families of cones, which include symmetric cones and the so-called smooth cones. We also give a family of examples showing that this bound can also fail to be sharp. In addition, we furnish a new proof of a result by G\"uler and Tun\c{c}el which states that the Carath\'eodory number of a symmetric cone is equal to its rank. Finally, we connect our discussion to the notion of cp-rank for completely positive matrices.