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Variance estimates and almost Euclidean structure

Grigoris Paouris and Petros Valettas
From the journal Advances in Geometry

Abstract

We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on ℝn, which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky’s theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.

  1. Communicated by: M. Henk

  2. Funding The first author was supported by the NSF CAREER-1151711 grant. The second author was supported by the NSF grant DMS-1612936.

Acknowledgement

The authors are grateful to Peter Pivovarov for useful remarks. They would also like to thank the anonymous referee whose helpful comments improved the presentation of the paper.

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Received: 2017-04-11
Published Online: 2019-04-09
Published in Print: 2019-04-24

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