On the entropy of Hilbert Geometries of Low Regularities

TL;DR

This paper investigates how the regularity of a convex set's boundary influences its Finslerian volume entropy, establishing a precise relationship in two-dimensional cases with Ahlfors regular curvature measures.

Contribution

It provides a novel explicit formula linking boundary regularity and volume entropy for 2D convex sets with Ahlfors regular curvature measures.

Findings

Volume entropy equals 2α/(α+1) for Ahlfors α-regular curvature measures.

Establishes a quantitative connection between boundary regularity and entropy.

Advances understanding of geometric properties in Hilbert geometries.

Abstract

We compare the regularity of the boundary of a convex set with the value of its Finslerian volume entropy. The main result states that the volume entropy of a two-dimensional domain whose associated curvature measure is Ahlfors $\alpha$-regular is $\frac{2\alpha}{\alpha +1}$.