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Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. (English) Zbl 0985.58019
Summary: The authors show that transport inequalities, similar to the one derived by M. Talagrand [Geom. Funct. Anal. 6, 587-600 (1996; Zbl 0859.46030)] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand’s inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately loc-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.

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
28A35 Measures and integrals in product spaces
60E15 Inequalities; stochastic orderings
60G15 Gaussian processes
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