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

 

A note on subgaussian estimates for linear functionals on convex bodies


Authors: A. Giannopoulos, A. Pajor and G. Paouris
Journal: Proc. Amer. Math. Soc. 135 (2007), 2599-2606
MSC (2000): Primary 52A20; Secondary 46B07
Published electronically: March 29, 2007
MathSciNet review: 2302581
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Abstract: We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $ K$ is a convex body in $ {\mathbb{R}}^n$ with volume one and center of mass at the origin, there exists $ x\neq 0$ such that

$\displaystyle \vert\{ y\in K:\,\vert\langle y,x\rangle \vert\geq t\Vert\langle\cdot ,x\rangle\Vert _1\}\vert\leq\exp (-ct^2/\log^2(t+1))$

for all $ t\geq 1$, where $ c>0$ is an absolute constant. The proof is based on the study of the $ L_q$-centroid bodies of $ K$. Analogous results hold true for general log-concave measures.


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Additional Information

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

A. Pajor
Affiliation: Équipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Champs sur Marne, 77454, Marne-la-Vallée, Cedex 2, France
Email: Alain.Pajor@univ-mlv.fr

G. Paouris
Affiliation: Équipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Champs sur Marne, 77454, Marne-la-Vallée, Cedex 2, France
Email: grigoris_paouris@yahoo.co.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08778-3
PII: S 0002-9939(07)08778-3
Keywords: Isotropic convex bodies, concentration of volume, tail estimates for linear functionals, $L_q$--centroid bodies
Received by editor(s): April 20, 2006
Published electronically: March 29, 2007
Additional Notes: The project was co-funded by the European Social Fund and National Resources - (EPEAEK II) “Pythagoras II". The second named author would like to thank the Department of Mathematics of the University of Athens for the hospitality. The third named author was supported by a Marie Curie Intra-European Fellowship (EIF), Contract MEIF-CT-2005-025017.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2007 American Mathematical Society