Volume entropy of Hilbert Geometries

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.