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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Uniform almost sub-Gaussian estimates for linear functionals on convex sets

Author: B. Klartag
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 77-106
MSC (2000): Primary 53A20, 52A21
Published electronically: December 17, 2007
MathSciNet review: 2319512
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Abstract: A well-known consequence of the Brunn-Minkowski inequality says that the distribution of a linear functional on a convex set has a uniformly subexponential tail. That is, for any dimension $ n$, any convex set $ K \subset \mathbb{R}^n$ of volume one, and any linear functional $ \varphi: \mathbb{R}^n \rightarrow \mathbb{R}$, we have

$\displaystyle \operatorname{Vol}_n (\lbrace x \in K; \vert\varphi(x)\vert > t \Vert \varphi \Vert _{L_1(K)}\rbrace) \le e^{- c t} \enskip$for all$\displaystyle \enskip t > 1, $

where $ \Vert \varphi \Vert _{L_1(K)} = \int_K \vert\varphi(x)\vert d x$ and $ c > 0$ is a universal constant. In this paper, it is proved that for any dimension $ n$ and a convex set $ K \subset \mathbb{R}^n$ of volume one, there exists a nonzero linear functional $ \varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that

$\displaystyle \operatorname{Vol}_n(\lbrace x \in K; \vert\varphi(x)\vert > t \Vert \varphi \Vert _{L_1(K)}\rbrace) \le e^{- c \frac{t^2}{\log^5 (t+1)} } \enskip$for all$\displaystyle \enskip t > 1, $

where $ c > 0$ is a universal constant.

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Additional Information

B. Klartag
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540

Keywords: Brunn--Minkowski inequality, convex body, logarithmic Laplace transform
Received by editor(s): August 1, 2006
Published electronically: December 17, 2007
Additional Notes: The author is a Clay Research Fellow and was also supported by NSF (grant no. DMS-0456590)
Dedicated: In honor of Professor V. A. Zalgaller
Article copyright: © Copyright 2007 American Mathematical Society

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