, Volume 16, Issue 5, pp 1021-1049
Date: 09 Nov 2006

Concentration of mass on convex bodies

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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 L K denotes the isotropic constant.

Research supported by a Marie Curie Intra-European Fellowship (EIF), Contract MEIF-CT-2005-025017. Part of this work was done while the author was a Postdoctoral Fellow at the University of Athens under the EPEAEK program “Pythagoras II”.
Received: January 2006; Revision: March 2006; Accepted: March 2006