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Small ball probability estimates for log-concave measures


Author: Grigoris Paouris
Journal: Trans. Amer. Math. Soc. 364 (2012), 287-308
MSC (2010): Primary 52A20; Secondary 46B07
DOI: https://doi.org/10.1090/S0002-9947-2011-05411-5
Published electronically: June 8, 2011
MathSciNet review: 2833584
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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$,

$\displaystyle \mathbb{P} \left( \Vert Ax-y\Vert _{2} \leq \varepsilon \Vert A\V... ..._{2}}{b}\frac{\Vert A\Vert _{{\rm HS}}}{\Vert A\Vert _{{\rm op}}} \big )^{2} },$

where $ c_{1}, c_{2}>0$ are absolute constants.


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Additional Information

Grigoris Paouris
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: grigoris_paouris@yahoo.co.uk

DOI: https://doi.org/10.1090/S0002-9947-2011-05411-5
Keywords: Log-concave probability measures, small ball probability estimates, isotropic constant
Received by editor(s): January 28, 2009
Received by editor(s) in revised form: May 31, 2010, and June 7, 2010
Published electronically: June 8, 2011
Additional Notes: The author is partially supported by an NSF grant (DMS 0906150)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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