The displacement convexity is the convexity of functionals defined on the

space of probability measures equipped with the distance function,

so-called Wasserstein distance function. In this talk, I first introduce a

class of generalized relative entropies, which stems from the Bregman

divergence, on a Riemannian manifold with a weighted measure. Then I prove

that the convexity of all the entropies in this class is equivalent to the

combination of the nonnegative weighted Ricci curvature and the convexity

of another function used in the definition of the generalized relative

entropies. From the convexity of the generalized relative entropies, I

derive appropriate variants of the Talagrand and the logarithmic Sobolev

inequalities. I also investigate the gradient flow of the gradient flow of

the generalized relative entropy. This is the joint work with Shin-ichi

Ohta.

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