Poincaré inequality for non euclidean metrics and transportation cost inequalities on ${\mathbb{R}}^d$

Abstract

In this paper, we consider Poincaré inequalities for non euclidean metrics on ${\mathbb{R}}^d$. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportation-cost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized Beckner-Latala-Oleszkiewicz inequalities.