Small ball probability estimates for log-concave measures

Paouris, Grigoris

doi:10.1090/S0002-9947-2011-05411-5

Small ball probability estimates for log-concave measures
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Trans. Amer. Math. Soc. 364 (2012), 287-308 Request permission

Abstract:

We establish a small ball probability inequality for isotropic $\log$-concave probability measures: there exist absolute constants $c_{1}, c_{2}>0$ such that if $X$ is an isotropic $\log$-concave random vector in ${\mathbb R}^n$ with $\psi _{2}$ constant and bounded by $b$ and if $A$ is a non-zero $n\times n$ matrix, then for every $\varepsilon \in (0,c_{1})$ and $y \in \mathbb R^n$, \[ \mathbb P \left ( \|Ax-y\|_{2} \leq \varepsilon \|A\|_{\textrm {HS}} \right ) \leq \varepsilon ^{\big ( \frac {c_{2}}{b}\frac {\|A\|_{\textrm {HS}}}{\|A\|_{\textrm {op}}} \big )^{2} },\] where $c_{1}, c_{2}>0$ are absolute constants.

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