Concentration and isoperimetry are equivalent assuming curvature lower bound

Emanuel Milman

doi:10.1016/j.crma.2008.11.014

Differential Geometry

Concentration and isoperimetry are equivalent assuming curvature lower bound
[Concentration et isopérimétrie sont équivalentes lorsque la courbure est bornée inférieurement]

Présenté par : Jean Bourgain

Emanuel Milman ¹

¹ School of Mathematics, Institute for Advanced Study, Einstein Drive, Simonyi Hall, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 73-76.

Résumé
Abstract

Il est connu que les inégalités isopérimétriques impliquent dans un cadre espace-mesure-métrique très général des inégalités de concentration correspondantes. Les premières bornent la mesure de bord d'un ensemble en fonction de sa mesure, tandis que les dernières bornent la mesure d'un ensemble qui est séparé d'un ensemble ayant la moitié de la mesure totale, en fonction de leur distance mutuelle. Nous montrons que sous une condition de borne inférieure sur le tenseur de courbure de Bakry–Émery d'une variété riemannienne équipée d'une densité, les inégalités de concentration complètement générales impliquent inversement leurs analogues isopérimétriques, à des constantes près qui sont indépendantes de la dimension. Notre méthode est entièrement géométrique, continuant l'approche de Gromov, Buser et Morgan.

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry–Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. Our method is entirely geometric, continuing the approach of Gromov, Buser and Morgan.

Reçu le : 2008-11-17
Accepté le : 2008-11-24
Publié le : 2008-12-24

DOI : 10.1016/j.crma.2008.11.014

Affiliations des auteurs :

Emanuel Milman ¹

¹ School of Mathematics, Institute for Advanced Study, Einstein Drive, Simonyi Hall, Princeton, NJ 08540, USA

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Emanuel Milman. Concentration and isoperimetry are equivalent assuming curvature lower bound. Comptes Rendus. Mathématique, Volume 347 (2009) no. 1-2, pp. 73-76. doi : 10.1016/j.crma.2008.11.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.11.014/

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