Concentration of mass on convex bodies

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.

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