Communications in Analysis and Geometry

Volume 21 (2013)

Number 4

Displacement convexity of generalized relative entropies. II

Pages: 687 – 785



Shin-Ichi Ohta (Department of Mathematics, Kyoto University, Kyoto, Japan)

Asuka Takatsu (Graduate School of Mathematics, Nagoya University, Nagoya, Japan)


We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the non-negative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani’s version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy.

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