Hyperbolicity cones are amenable
Abstract
Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or "nice") which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.
Cite
@article{arxiv.2102.06359,
title = {Hyperbolicity cones are amenable},
author = {Bruno F. Lourenço and Vera Roshchina and James Saunderson},
journal= {arXiv preprint arXiv:2102.06359},
year = {2023}
}
Comments
11 pages. v2: added Section 3.2 giving an explicit description of the span of a face and Section 3.3 giving concrete examples. Minor edits throughout