Approximability of convex bodies and volume entropy in Hilbert geometry

TL;DR

This paper explores the relationship between the approximability of convex bodies and volume entropy in Hilbert geometry, establishing key equalities and bounds in low dimensions and solving a longstanding conjecture.

Contribution

It proves that twice the approximability equals volume entropy in dimensions two and three, and provides bounds and proofs for higher dimensions.

Findings

In dimension two, entropy equals twice the approximability.

In dimension three, entropy is bounded below by twice the approximability.

Solved the entropy upper bound conjecture in dimension three.

Abstract

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).