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Algebra i Analiz, 2007, Volume 19, Issue 1, Pages 109–148 (Mi aa105)  

This article is cited in 20 scientific papers (total in 20 papers)

Research Papers

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

B. Klartag

School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA
References:
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 $\varphi\colon\mathbb{R}^n\to\mathbb{R}$, we have
$$ \operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-ct}\quad \text{for all}\quad t>1, $$
where $\|\varphi\|_{L_1(K)}=\int_K|\varphi(x)|\,dx$ 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 $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ such that
$$ \operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-c\frac{t^2}{\log^5 (t+1)}} \quad \text{for all}\quad t>1, $$
where $c>0$ is a universal constant.
Received: 01.08.2006
English version:
St. Petersburg Mathematical Journal, 2008, Volume 19, Issue 1, Pages 77–106
DOI: https://doi.org/10.1090/S1061-0022-07-00987-9
Bibliographic databases:
Document Type: Article
MSC: 3A20, 52A21
Language: English
Citation: B. Klartag, “Uniform almost sub-gaussian estimates for linear functionals on convex sets”, Algebra i Analiz, 19:1 (2007), 109–148; St. Petersburg Math. J., 19:1 (2008), 77–106
Citation in format AMSBIB
\Bibitem{Kla07}
\by B.~Klartag
\paper Uniform almost sub-gaussian estimates for linear functionals on convex sets
\jour Algebra i Analiz
\yr 2007
\vol 19
\issue 1
\pages 109--148
\mathnet{http://mi.mathnet.ru/aa105}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2319512}
\zmath{https://zbmath.org/?q=an:1140.60010}
\transl
\jour St. Petersburg Math. J.
\yr 2008
\vol 19
\issue 1
\pages 77--106
\crossref{https://doi.org/10.1090/S1061-0022-07-00987-9}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000267653000006}
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  • https://www.mathnet.ru/eng/aa/v19/i1/p109
  • This publication is cited in the following 20 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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