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A Short Proof of Paouris' Inequality

Published online by Cambridge University Press:  20 November 2018

Radosław Adamczak
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.plrlatala@mimuw.edu.pl
Rafał Latała
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.plrlatala@mimuw.edu.pl
Alexander E. Litvak
Affiliation:
Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: alexandr@math.ualberta.cakoles@mimuw.edu.pl
Krzysztof Oleszkiewicz
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland e-mail: radamcz@mimuw.edu.plrlatala@mimuw.edu.pl
Alain Pajor
Affiliation:
Université Paris-Est, Équipe d'Analyse et Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France e-mail: nicole.tomczak@ualberta.caAlain.Pajor@univ-mlv.fr
Nicole Tomczak-Jaegermann
Affiliation:
Department of Mathematics and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: alexandr@math.ualberta.cakoles@mimuw.edu.pl
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Abstract

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We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$,

$$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$

More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$,

$${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle \right|}^{p}})}^{1/p}}.$$

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

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