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

DOI:
https://doi.org/10.1090/S0002-9939-07-08778-3

Published electronically:
March 29, 2007

MathSciNet review:
2302581

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 \[ |\{ y\in K: |\langle y,x\rangle |\geq t\|\langle \cdot ,x\rangle \|_1\}|\leq \exp (-ct^2/\log ^2(t+1))\] 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

MR Author ID:
671202

Email:
grigoris_paouris@yahoo.co.uk

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