On the Hilbert Geometry of Convex Polytopes

TL;DR

This paper surveys the Hilbert geometry of convex polytopes, focusing on characterizations based on volume growth of metric balls and bi-Lipschitz equivalence to simplex geometry.

Contribution

It provides two key characterizations of Hilbert geometries of convex polytopes, enhancing understanding of their metric and geometric properties.

Findings

Volume growth of metric balls characterizes Hilbert geometries of polytopes.

Hilbert geometry of polytopes is bi-Lipschitz equivalent to simplex geometry.

Abstract

We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.