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Dario Cordero-Erausquin; Robert J. McCann; Michael Schmuckenschläger
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
Annales de la faculté des sciences de Toulouse Sér. 6, 15 no. 4 (2006), p. 613-635, doi: 10.5802/afst.1132
Article PDF | Reviews MR 2295207 | Zbl 1125.58007

Résumé - Abstract

We investigate Prékopa-Leindler type inequalities on a Riemannian manifold $M$ equipped with a measure with density $e^{-V}$ where the potential $V$ and the Ricci curvature satisfy $\operatorname{Hess}_x V + \operatorname{Ric}_x \ge \lambda \, I$ for all $x\in M$, with some $\lambda \in \mathbb{R}$. As in our earlier work [14], the argument uses optimal mass transport on $M$, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.

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