Weighted Ricci curvature estimates for Hilbert and Funk geometries

Shin-ichi Ohta

Vol. 265 (2013), No. 1, 185–197

DOI: 10.2140/pjm.2013.265.185

Abstract

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

Keywords

Hilbert geometry, Funk geometry, Ricci curvature, curvature-dimension condition

Mathematical Subject Classification 2010

Primary: 53C60

Secondary: 53C23

Milestones

Received: 24 July 2012

Accepted: 16 October 2012

Published: 28 August 2013