Volume entropy of Hilbert Geometries

Details Volume entropy of Hilbert Geometries Volume entropy of Hilbert Geometries

2008-10-07

arxiv.org/abs/0810.1123v2

Pacific J. Math., 245 (2010), 201-225

Mathematics

Differential Geometry

27 pages; minor changes

Scientific paper

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.

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