Some applications of mass transport to Gaussian-type inequalities

As discovered by Brenier, mapping through a convex gradient gives the optimal transport in R-n. In the present article, this map is used in the setting of Gaussian-like measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.