From dimension free concentration to Poincar\'e inequality

TL;DR

This paper establishes an equivalence between dimension-free concentration inequalities and the Poincaré inequality for probability measures on metric spaces, deepening understanding of measure concentration phenomena.

Contribution

It proves that a probability measure satisfies a non-trivial dimension-free concentration inequality for the _2 metric if and only if it satisfies the Poincare9 inequality, linking two fundamental concepts.

Findings

Dimension-free concentration is equivalent to the Poincare9 inequality.

Provides a characterization of measures satisfying concentration inequalities.

Bridges measure concentration and functional inequalities in metric spaces.

Abstract

We prove that a probability measure on an abstract metric space satisfies a non trivial dimension free concentration inequality for the $\ell_2$ metric if and only if it satisfies the Poincar\'e inequality.