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.