$\psi _{\alpha }$estimates for marginals of logconcave probability measures
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 by A. Giannopoulos, G. Paouris and P. Valettas
 Proc. Amer. Math. Soc. 140 (2012), 12971308
 DOI: https://doi.org/10.1090/S000299392011109845
 Published electronically: August 3, 2011
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Abstract:
We show that a random marginal $\pi _F(\mu )$ of an isotropic logconcave 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, superGaussian.

[(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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Bibliographic 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
 MR Author ID: 671202
 Email: grigoris_paouris@yahoo.co.uk
 P. Valettas
 Affiliation: Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece
 MR Author ID: 957443
 Email: petvalet@math.uoa.gr
 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
 © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.  Journal: Proc. Amer. Math. Soc. 140 (2012), 12971308
 MSC (2010): Primary 46B07; Secondary 52A20
 DOI: https://doi.org/10.1090/S000299392011109845
 MathSciNet review: 2869113