Volume entropy of Hilbert Geometries

TL;DR

This paper proves that the volume entropy of Hilbert geometries for smooth convex bodies in n-dimensions equals n-1, introduces a new projective invariant, and explores entropy bounds and examples in 2D.

Contribution

It establishes the exact volume entropy for smooth convex bodies and introduces a novel projective invariant related to convex geometry.

Findings

Volume entropy equals n-1 for $C^{1,1}$ convex bodies in n-dimensions.

In 2D, entropy is bounded above by a function of the Minkowski dimension.

Constructs an example of a plane Hilbert geometry with entropy between 0 and 1.

Abstract

It is shown that the volume entropy of a Hilbert geometry associated to an $n$-dimensional convex body of class $C^{1,1}$ equals $n-1$. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed. In the case $n=2$, and without any assumption on the boundary, it is shown that the entropy is bounded above by $\frac{2}{3-d} \leq 1$, where $d$ is the Minkowski dimension of the extremal set of $K$. An example of a plane Hilbert geometry with entropy strictly between 0 and 1 is constructed.