Lipschitz characterisation of polytopal Hilbert geometries

Abstract

We study Hilbert geometries admitting similar singularities on their boundary to those of a simplex. We show that in an adapted neighborhood of those singularities, two such geometries are bi-Lipschitz. As a corollary we prove that the Hilbert geometry of a convex set is bi-Lipschitz equivalent to a normed vector space if and only if the convex is a polytope.