On Homogeneous Convex Cones, The Carathéodory Number, and the Duality Mapping

Using three simple examples, we answer three questions related to homogeneous convex cones, the Carathéodory 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 Carathéodory number in the interior. Third, we show that the invariance of the Carathéodory 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 Carathéodory number of epigraphs of matrix norms.