2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69451We show that the quadratic transportation cost inequality $T_2$ is equivalent to both a Poincaré 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 Poincaré inequality is satisfied.ProbabilityFunctional Analysis47D07;60E15;60G10text