# On the convex Poincar\'e inequality and weak transportation inequalities

## TL;DR

This paper establishes an equivalence between the convex Poincaré inequality and weak transportation inequalities with quadratic-linear cost for probability measures on R^n, extending previous results and introducing new concentration inequalities.

## Contribution

It generalizes the equivalence between convex Poincare9 and weak transportation inequalities to higher dimensions and introduces modified logarithmic Sobolev inequalities for convex functions.

## Key findings

- Proves the equivalence for R^n
- Derives refined concentration inequalities for convex functions
- Extends previous one-dimensional results

## Abstract

We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar and Tetali.