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 J. Approx. Theory, 2013, Volume 175, Pages 77–82 (Mi jath4)

On the concentration of measure and the $L^1$-norm

Yu. V. Malykhina, K. S. Ryutinb

a Steklov Mathematical Institute, Gubkina str. 8, 119991, Moscow, Russia
b MSU, Faculty of Mechanics and Mathematics, GSP-1, 1 Leninskiye Gory, Main Building, 119991, Moscow, Russia

Abstract: We seek a function in an $N$-dimensional subspace of $L^1$ that attains a half (more generally, a given part) of its $L^1$-norm on a set of least possible measure. We prove that such a measure is asymptotically maximal when the space is spanned by independent standard normal variables. This answers a question of Y. Benyamini, A. Kroó, A. Pinkus.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00329 Ministry of Education and Science of the Russian Federation Nsh-6431.2012.1Nsh-6003.2012.1 The authors were supported by the RBFR, grant 11-01-00329 (both authors) and the Programme for support of leading scientific schools, grants Nsh-6431.2012.1 (first author) and Nsh-6003.2012.1 (second author).

DOI: https://doi.org/10.1016/j.jat.2013.07.011

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Accepted:21.07.2013
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