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On the volume of caps and bounding the mean-width of an isotropic convex body

Published online by Cambridge University Press:  10 May 2010

PETER PIVOVAROV
PETER PIVOVAROV*
Affiliation:
Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, Canada, T6G-2G1. e-mail: ppivovarov@math.ualberta.ca
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Abstract

Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 317 - 331
Copyright
Copyright © Cambridge Philosophical Society 2010

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