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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \psi_{\alpha }$-estimates for marginals of log-concave probability measures


Authors: A. Giannopoulos, G. Paouris and P. Valettas
Journal: Proc. Amer. Math. Soc. 140 (2012), 1297-1308
MSC (2010): Primary 46B07; Secondary 52A20
Published electronically: August 3, 2011
MathSciNet review: 2869113
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Abstract: We show that a random marginal $ \pi_F(\mu )$ of an isotropic log-concave probability measure $ \mu$ on $ \mathbb{R}^n$ exhibits better $ \psi_{\alpha }$-behavior. For a natural variant $ \psi_{\alpha }^{\prime }$ of the standard $ \psi_{\alpha }$-norm we show the following:

(i)
If $ k\leq\sqrt{n}$, then for a random $ F\in G_{n,k}$ we have that $ \pi_F(\mu )$ is a $ \psi_2^{\prime }$-measure. We complement this result by showing that a random $ \pi_F(\mu )$ is, at the same time, super-Gaussian.
(ii)
If $ k=n^{\delta }$, $ \frac{1}{2}<\delta <1$, then for a random $ F\in G_{n,k}$ we have that $ \pi_F(\mu )$ is a $ \psi_{\alpha (\delta )}^{\prime }$-measure, where $ \alpha (\delta )=\frac{2\delta }{3\delta -1}$.


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Additional Information

A. Giannopoulos
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
Email: apgiannop@math.uoa.gr

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

P. Valettas
Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
Email: petvalet@math.uoa.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10984-5
PII: S 0002-9939(2011)10984-5
Keywords: Log-concave probability measures, random marginals, isotropic constant
Received by editor(s): July 27, 2010
Received by editor(s) in revised form: December 24, 2010
Published electronically: August 3, 2011
Additional Notes: The second author was partially supported by an NSF grant
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.