Carathéodory number of homogeneous convex cones

2025/11/21 by Chek Beng Chua, Chua, Chek Beng
Computer Science · Mathematics · #90C25 (Primary) 52A20 (Secondary) #Advanced Graph Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2511.17051

openalex publication_date 2025/11/21 · openalex created_date 2025/11/25 · openalex updated_date 2026/07/28

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.

Citations

• Facial structure of copositive and completely positive cones over a second-order cone
• Faces of homogeneous cones and applications to homogeneous chordality
• A note on faces of convex sets
• Linear optimization over homogeneous matrix cones
• A bound on the Carathéodory number

Related

• Faces of homogeneous cones and applications to homogeneous chordality
• A bound on the Carathéodory number
• Linear optimization over homogeneous matrix cones
• Minimal hyperbolic polynomials and ranks of homogeneous cones
• An example of homogeneous cones whose basic relative invariant has maximal degree