VOLUME ENTROPY OF HILBERT GEOMETRIES

We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 - d) <= 1, where d is the Minkowski dimension of the extremal set of K, and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area.