Title:
Geometry of Homogeneous Convex Cones,
Duality Mapping, and Optimal Self-Concordant Barriers

Speaker: Levent Tuncel

Abstract:
I will discuss homogeneous convex cones. We first characterize
the extreme rays of such cones in the context of their primal
construction (due to Vinberg) and also in the context of their
dual construction (due to Rothaus). Then, using these results,
we prove that every homogeneous cone is facially exposed. We provide
an alternative proof of a result of G\"uler and Tun\c{c}el that the Siegel
rank of a symmetric cone is equal to its Carath\'eodory number.
Our proof does not use the Jordan-vonNeumann-Wigner
characterization of the symmetric cones but it easily follows
from the primal construction of the homogeneous cones and
our results on the geometry of homogeneous cones in primal and dual forms.
 We briefly discuss the duality mapping in the context
of automorphisms of convex cones and prove that the duality mapping is
not an involution on certain self-dual cones.

This is based on joint work with Van Anh Truong.