Lipschitz Characterisation of Polytopal Hilbert Geometries

TL;DR

This paper establishes that Hilbert geometries are bi-Lipschitz equivalent to normed vector spaces precisely when the underlying convex set is a polytope, providing a clear geometric characterization.

Contribution

It provides a complete characterization of polytopal Hilbert geometries via bi-Lipschitz equivalence to normed spaces.

Findings

Hilbert geometry of a convex set is bi-Lipschitz to a normed space iff the set is a polytope.

Characterization of polytopal Hilbert geometries.

Connection between convex geometry and metric space theory.

Abstract

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.