Equivalence of the Poincar\'e inequality with a transport-chi-square inequality in dimension one

TL;DR

This paper establishes that in one dimension, the Poincaré inequality is equivalent to a new transport-chi-square inequality involving Wasserstein and chi-square distances, and confirms its tensorization property.

Contribution

It introduces a novel transport-chi-square inequality in dimension one and proves its equivalence to the Poincaré inequality, including tensorization properties.

Findings

Equivalence between Poincaré and transport-chi-square inequalities in 1D

Tensorization property of the new inequality

Linking Wasserstein distance with chi-square pseudo-distance

Abstract

In this paper, we prove that, in dimension one, the Poincar\'e inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance. We also check tensorization of this transport-chi-square inequality.