GENERALIZATION OF AN INEQUALITY BY TALAGRAND AND LINKS WITH THE LOGARITHMIC SOBOLEV INEQUALITY

GENERALIZATION OF AN INEQUALITY BY TALAGRAND, AND LINKS WITH THE LOGARITHMIC SOBOLEV INEQUALITY F. OTTO AND C. VILLANI Abstract. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are im- plied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial differential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 10 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 29 7. Linearizations 31 Appendix A. A nonlinear approximation argument 34 References 35 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance (1) d(x, y) = inf ? ? ? √∫ 1 0 |w˙(t)|2 dt, w ? C1((0, 1);M), w(0) = x, w(1) = y ? ? ? . 1

• dµ d?


• measure ?


• d?


• sobolev inequality


• gross's logarithmic


• gotze also


• d? ?


• gaussian measure