Flag-approximability of convex bodies and volume growth of Hilbert geometries

Constantin Vernicos (1) , Cormac Walsh (2, 3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 TROPICAL - TROPICAL
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We show that the volume entropy of a Hilbert geometry on a convex body is exactly twice the flag-approximability of the body. We then show that both of these quantities are maximized in the case of the Euclidean ball. We also compute explicitly the asymptotic volume of a convex polytope, which allows us to prove that simplices have the least asymptotic volume, as was conjectured by the first author.

Domaines: Géométrie métrique [math.MG]

Constantin Vernicos, Cormac Walsh. Flag-approximability of convex bodies and volume growth of Hilbert geometries. Annales Scientifiques de l'École Normale Supérieure, 2021, 54, pp.1297-1315. ⟨10.24033/asens.2482⟩. ⟨hal-01423693⟩