Geometric & Functional Analysis GAFA

, Volume 16, Issue 5, pp 1021–1049 | Cite as

Concentration of mass on convex bodies

Article

Abstract.

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 bodies concentration of volume tail estimates for linear functionals Lq-centroid bodies 

AMS Mathematics Subject Classification.

Primary 52A20 Secondary 46B07 

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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

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