Approximability of convex bodies and volume entropy of Hilbert geometries. preprint 2012. Institut de mathématique et de modélisation de

Abstract. The approximability of a convex body is a number which measures the difficulty to approximate that body by poly-topes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in di-mension three and give a new proof in dimension two from the one found in [BBV10]. Introduction and statement of results Hilbert geometries are a family of metric spaces defined in the in-terior of an open and bounded convex set using cross-ratios following the construction of the hyperbolic geometry’s projective model [Hil71]. They are actually length space with an underlying Finsler structur