On the concentration of measure and the \(L^1\)-norm
type: Item
instance of: scholarly article
title: On the concentration of measure and the \(L^1\)-norm
zbMATH Open document ID: 1298.41019
author: K. S. Ryutin
author: Yuri V. Malykhin
published in: Journal of Approximation Theory
publication date: 2014-09-10T00:00:00Z
review text: Introducing a certain quantity \(\alpha ^* (M)\), where \(M\subset L^1 =: L^1 (\Xi ,\mu) \) with \( (\Xi ,\mu) \) as some measurable space. \textit{Y. Benyamini} et al. [Constr. Approx. 36, No. 3, 399--431 (2012; Zbl 1262.41017), p. 246] proposed a conjecture on the maximal value of \(\alpha^*\) over all infinite-dimensional subspaces of \(L^1\). In the present paper the authors decide this conjecture. The proof presented here is based on the concentration of measure phenomenon (see e.g. [\textit{M. Ledoux}, The concentration of measure phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0995.60002)]).
reviewed by: H. P. Dikshit
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 6342397
zbMATH Keywords: concentration of measure
zbMATH Keywords: \(L^1\) approximation
zbMATH Keywords: minimal support
MaRDI profile type: Publication
full work available at URL: j.jat.2013.07.011
OpenAlex ID: W2030431328
cites work: Q4387224
cites work: \(L^{1}\)-approximation and finding solutions with small support
cites work: On concentration of distributions of random weighted sums
cites work: The concentration of measure phenomenon
cites work: Asymptotic theory of finite dimensional normed spaces. With an appendix by M. Gromov: Isoperimetric inequalities in Riemannian manifolds
DOI: 10.1016/J.JAT.2013.07.011
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description: scientific article; zbMATH DE number 6342397