Carathéodory number of homogeneous convex cones
Chek Beng Chua
TL;DR
This work studies the Carathéodory number κ_{ar{K}} of homogeneous convex cones through their spectrahedral representations, linking κ_{ar{K}} to the cone's rank and to selfduality. Using Ishi's block matrix spectrahedral representations, the authors derive explicit facial structures for the closures and duals, establish operators that relate block components, and obtain necessary and sufficient conditions under which the rank matches Carathéodory numbers for both the closure and the dual. A key result is that a homogeneous cone is selfdual if and only if its rank matches the Carathéodory numbers of both its closure and its dual, with indecomposable components characterizing this property; the sparse-cone case is then resolved by a graph-theoretic criterion: homogeneous sparse spectrahedral cones correspond precisely to homogeneous chordal graphs. The findings provide geometric and algebraic criteria for equality of these invariants, with implications for conic optimization and the structure of sparse matrix representations. The methods unify spectrahedral geometry, facial theory, and graph sparsity patterns to illuminate when rank-based bounds are tight and how selfduality arises in this class.
Abstract
We study the Carathéodory number of homogeneous convex cones via their spectrahedral representations. A characterization of homogeneous convex cones whose ranks match their Carathéodory 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 Carathéodory 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.