Volume entropy of Hilbert geometries

Gautier Berck, Andreas Bernig, Constantin Vernicos
2010 Pacific Journal of Mathematics  
It is shown that among all plane Hilbert geometries, the hyperbolic plane has the maximal volume entropy. More precisely, it is shown that the volume entropy is bounded above by 2 3−d ≤ 1, where d is the Minkowski dimension of the extremal set of K. An explicit example of a plane Hilbert geometry with noninteger volume entropy is constructed. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve
more » ... thesis. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed.
doi:10.2140/pjm.2010.245.201 fatcat:dtao6bci3rht5hjnd55omzurju