From dimension free concentration to Poincaré inequality

Nathael Gozlan (1) , Cyril Roberto (2) , Paul-Marie Samson (1)

1 Laboratoire d'Analyse et de Mathématiques Appliquées
2 MODAL'X - Modélisation aléatoire de Paris X

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 Poincaré inequality.

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Nathael Gozlan, Cyril Roberto, Paul-Marie Samson. From dimension free concentration to Poincaré inequality. Calculus of Variations and Partial Differential Equations, 2015. ⟨hal-00823733v2⟩