On the Hilbert Geometry of products

TL;DR

This paper investigates the properties of Hilbert geometries on product convex sets, establishing bi-Lipschitz equivalence, additive volume entropy, and conditions for amenability based on the factors.

Contribution

It proves that the Hilbert geometry of a product of convex sets is bi-Lipschitz equivalent to the product of their geometries, and shows volume entropy and amenability are additive or equivalent.

Findings

Hilbert geometry of product convex sets is bi-Lipschitz equivalent to the product of individual geometries

Volume entropy is additive under product operations

Amenability of a product equals the amenability of each factor

Abstract

We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that amenability of a product is equivalent to the amenability of each terms.