Displacement convexity of generalized relative entropies. II

Shin Ichi Ohta, Asuka Takatsu

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)689-785
Number of pages97
JournalCommunications in Analysis and Geometry
Issue number4
Publication statusPublished - 2013
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Geometry and Topology
  • Statistics, Probability and Uncertainty


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