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

TL;DR

This paper introduces the concept of flag-approximability for convex bodies, linking it to volume entropy in Hilbert geometries, and provides explicit volume calculations highlighting extremal cases like Euclidean balls and simplices.

Contribution

It establishes a precise relationship between flag-approximability and volume entropy, and computes asymptotic volumes for convex polytopes, revealing extremal geometric properties.

Findings

Flag-approximability equals half the volume entropy of Hilbert geometry.

Euclidean balls maximize both flag-approximability and volume entropy.

Simplices have the least asymptotic volume among convex polytopes.

Abstract

We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both quantities are maximized when the convex body is a 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.