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Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 19, Issue 2

# Variance estimates and almost Euclidean structure

Grigoris Paouris
/ Petros Valettas
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/advgeom-2018-0030

## 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.

MSC 2010: Primary 46B07; 46B09; Secondary 52A21; 52A23

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Published Online: 2019-04-09

Published in Print: 2019-04-24

Communicated by: M. Henk

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

Citation Information: Advances in Geometry, Volume 19, Issue 2, Pages 165–189, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X,

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