Concentration of mass on isotropic convex bodies

Paouris, Grigoris

doi:10.1016/j.crma.2005.11.018

Functional Analysis

Concentration of mass on isotropic convex bodies
[Concentration de masse pour les corps convexes isotropes]

Présenté par : Gilles Pisier

Paouris, Grigoris ¹

¹ Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece

Comptes Rendus. Mathématique, Tome 342 (2006) no. 3, pp. 179-182.

Résumé
Abstract

Nous démontrons qu'il existe une constante absolue $c > 0$ , telle que, si K est un corps convexe isotrope, alors

Prob ({x \in K : ‖ x ‖_{2} ⩾ c \sqrt{n} L_{K} t}) ⩽ \exp (- \sqrt{n} t)

pour tout

t ⩾ 1

, où

L_{K}

désigne la constante d'isotropie.

We establish sharp concentration of mass for isotropic convex bodies: there exists an absolute constant $c > 0$ such that if K is an isotropic convex body in $R^{n}$ , then

Prob ({x \in K : ‖ x ‖_{2} ⩾ c \sqrt{n} L_{K} t}) ⩽ \exp (- \sqrt{n} t)

for every

t ⩾ 1

, where

L_{K}

denotes the isotropic constant.

Reçu le : 2005-11-09
Accepté le : 2005-11-16
Publié le : 2005-12-20

DOI : 10.1016/j.crma.2005.11.018

Affiliations des auteurs :

Paouris, Grigoris ¹

¹ Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece

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Paouris, Grigoris. Concentration of mass on isotropic convex bodies. Comptes Rendus. Mathématique, Tome 342 (2006) no. 3, pp. 179-182. doi : 10.1016/j.crma.2005.11.018. http://www.numdam.org/articles/10.1016/j.crma.2005.11.018/

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