Approximability of convex bodies and volume entropy of Hilbert geometries

Constantin Vernicos (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. 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 dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).

Domaines: Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]; Géométrie symplectique [math.SG]

Constantin Vernicos. Approximability of convex bodies and volume entropy of Hilbert geometries. Pacific Journal of Mathematics, 2017, 287 (1), pp.223-256. ⟨10.2140/pjm.2017.287.223⟩. ⟨hal-00819134⟩