Abstract
As discovered by Brenier, mapping through a convex gradient gives the optimal transport in ℝn. 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.
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Accepted December 15, 2000¶Published online January 28, 2002
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Cordero-Erausquin, D. Some Applications of Mass Transport to Gaussian-Type Inequalities. Arch. Rational Mech. Anal. 161, 257–269 (2002). https://doi.org/10.1007/s002050100185
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DOI: https://doi.org/10.1007/s002050100185