On the convex Poincare inequality and weak transportation inequalities

摘要

We prove that for a probability measure on R-n, the Poincare inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan, Roberto, Samson, Shu, Tetali and Feldheim, Marsiglietti, Nayar, Wang, concerning probability measures on the real line.