Geometric & Functional Analysis GAFA

, Volume 16, Issue 5, pp 1021–1049

Concentration of mass on convex bodies


DOI: 10.1007/s00039-006-0584-5

Cite this article as:
Paouris, G. GAFA, Geom. funct. anal. (2006) 16: 1021. doi:10.1007/s00039-006-0584-5


We establish sharp concentration of mass inequality for isotropic convex bodies: there exists an absolute constant c >  0 such that if K is an isotropic convex body in \(\mathbb{R}^{n}\), then
$$ {\text{Prob}}{\left( {{\left\{ {x \in K:||x||_{2} \geqslant c{\sqrt{n}}L_{K} t} \right\}}} \right)} \leqslant \exp {\left( { - {\sqrt{n}}t} \right)} $$
for every \(t\geqslant 1\), where LK denotes the isotropic constant.

Keywords and phrases.

Isotropic convex bodiesconcentration of volumetail estimates for linear functionalsLq-centroid bodies

AMS Mathematics Subject Classification.

Primary 52A20Secondary 46B07

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Équipe d’Analyse et de Mathématiques AppliquéesUniversité de Marne-la-ValléeMarne-la-Vallée, Cedex 2France