A criterion for Talagrand's quadratic transportation cost inequality
Abstract
We show that the quadratic transportation cost inequality T2 is equivalent to both a Poincar\'e inequality and a strong form of the Gaussian concentration property. The main ingredient in the proof is a new family of inequalities, called modified quadratic transportation cost inequalities in the spirit of the modified logarithmic-Sobolev inequalities by Bobkov and Ledoux BL97, that are shown to hold as soon as a Poincar\'e inequality is satisfied.
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