Asymptotic volume in Hilbert Geometries

TL;DR

This paper investigates the volume growth of metric balls in Hilbert geometries, establishing polynomial growth bounds and characterizing convex polytopes by their precise polynomial volume growth degree.

Contribution

It proves a lower bound on volume growth in Hilbert geometries and characterizes convex polytopes through their exact polynomial volume growth degree.

Findings

Volume growth in Hilbert geometries is at least polynomial of degree equal to the dimension.

Convex polytopes are characterized by having exactly polynomial volume growth of degree equal to the dimension.

Provides a geometric criterion for identifying convex polytopes based on volume growth.

Abstract

We prove that the metric balls of a Hilbert geometry admit a volume growth at least polynomial of degree their dimension. We also characterise the convex polytopes as those having exactly polynomial volume growth of degree their dimension.