Weighted Ricci curvature estimates for Hilbert and Funk geometries

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.Comment: 12 page

Nonlinear geometric analysis on Finsler manifolds

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the $\Gamma$-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.Comment: 37 pages, to appear in a topical issue of European Journal of Mathematics "Finsler Geometry: New Methods and Perspectives". arXiv admin note: text overlap with arXiv:1602.0039

Displacement convexity of generalized relative entropies

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the $m$-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.Comment: 43page