On homogeneous convex cones, Caratheodory number, and duality mapping

Levent Tuncel and Song Xu

Using three simple examples, we answer three questions related to homogeneous convex cones, the Carath{\'e}odory number of convex cones and self-concordant barriers for convex cones. First, we show that if the convex cone is not homogeneous then the duality mapping does not have to be an involution. Next we show that there are very elementary convex cones that are not homogeneous, but have invariant Carath{\'e}odory number in the interior. Third, we show that the invariance of the Carath{\'e}odory number in the interior of the convex cone does not suffice to make the cone homogeneous even if the cone is self-dual. Finally, we characterize the Carath{\'e}odory number of epi-graph of matrix norms.

CORR 99-21, Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1 Canada, June 1999.

Contact: ltuncel@math.uwaterloo.ca