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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

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

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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Uniform almost sub-Gaussian estimates for linear functionals on convex sets
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by B. Klartag
St. Petersburg Math. J. 19 (2008), 77-106
DOI: https://doi.org/10.1090/S1061-0022-07-00987-9
Published electronically: December 17, 2007

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 \begin{equation*} \operatorname {Vol}_n (\lbrace x \in K; |\varphi (x)| > t \| \varphi \|_{L_1(K)}\rbrace ) \le e^{- c t} \quad \text {for all}\quad t > 1, \end{equation*} where $\| \varphi \|_{L_1(K)} = \int _K |\varphi (x)| d x$ 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 \begin{equation*} \operatorname {Vol}_n(\lbrace x \in K; |\varphi (x)| > t \| \varphi \|_{L_1(K)}\rbrace ) \le e^{- c \frac {t^2}{\log ^5 (t+1)} } \quad \text {for all}\quad t > 1, \end{equation*} where $c > 0$ is a universal constant.
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Bibliographic Information
  • B. Klartag
  • Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
  • MR Author ID: 671208
  • Email: bklartag@math.princeton.edu
  • 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
  • © Copyright 2007 American Mathematical Society
  • Journal: St. Petersburg Math. J. 19 (2008), 77-106
  • MSC (2000): Primary 53A20, 52A21
  • DOI: https://doi.org/10.1090/S1061-0022-07-00987-9
  • MathSciNet review: 2319512