1,146 research outputs found
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 -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 -relative entropy, which stems from the Bregman
divergence, on weighted Riemannian and Finsler manifolds. We prove that the
displacement -convexity of the -relative entropy is equivalent to the
combination of the nonnegativity of the weighted Ricci curvature and the
-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
-relative entropy produces a solution to the porous medium equation or the
fast diffusion equation.Comment: 43page
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