Weighted Ricci curvature estimates for Hilbert and Funk geometries

TL;DR

This paper investigates the weighted Ricci curvature of Hilbert and Funk geometries on convex domains, establishing bounds and conditions that connect these geometries to curvature-dimension frameworks in metric measure space theory.

Contribution

It provides the first estimates of weighted Ricci curvature for Hilbert and Funk geometries, linking them to curvature-dimension conditions in a rigorous way.

Findings

Hilbert geometry has bounded weighted Ricci curvature.

Funk geometry has constant negative weighted Ricci curvature.

Both geometries satisfy the curvature-dimension condition.

Abstract

We consider Hilbert and Funk geometries on a strongly convex domain in the Euclidean space. We show that, with respect to the Lebesgue measure on the domain, Hilbert (resp. Funk) metric has the bounded (resp. constant negative) weighted Ricci curvature. As one of corollaries, these metric measure spaces satisfy the curvature-dimension condition in the sense of Lott, Sturm and Villani.