Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

We show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6, 587600) 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 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