Volume entropy of Hilbert geometries

Gautier Berck, Andreas Bernig and Constantin Vernicos

Vol. 245 (2010), No. 2, 201–225

DOI: 10.2140/pjm.2010.245.201

Abstract

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.

Keywords

metric geometry, Hilbert geometry, convex geometry

Mathematical Subject Classification 2000

Primary: 51F99, 53A20, 53C60

Milestones

Received: 12 March 2009

Revised: 24 September 2009

Accepted: 24 September 2009

Published: 1 April 2010