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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
Аннотация:
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.
Поступила в редакцию: 27.02.2013 Принята в печать: 21.07.2013
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