A criterion for Talagrand's quadratic transportation cost inequality

We show that the quadratic transportation cost inequality $T_2$ 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 \cite{BL97}, that are shown to hold as soon as a Poincar'e inequality is satisfied.