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
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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 \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.References
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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)
- © 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
Dedicated: In honor of Professor V. A. Zalgaller