$\psi _{\alpha }$estimates for marginals of logconcave probability measures
Authors:
A. Giannopoulos, G. Paouris and P. Valettas
Journal:
Proc. Amer. Math. Soc. 140 (2012), 12971308
MSC (2010):
Primary 46B07; Secondary 52A20
DOI:
https://doi.org/10.1090/S000299392011109845
Published electronically:
August 3, 2011
MathSciNet review:
2869113
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
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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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
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
Keywords:
Logconcave 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.