A remark on vertex index of the convex bodies

Efim D. Gluskin, Alexander E. Litvak

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

The vertex index of a symmetric convex body K Rn, vein(K), was introduced in [Bezdek, Litvak, Adv. Math. 215, 626-641 (2007)]. Bounds on the vertex index were given in the general case as well as for some basic examples. In this note we improve these bounds and discuss their sharpness. We show that which is asymptotically sharp. We also show that the estimate obtained in [Bezdek, Litvak, Adv. Math. 215, 626-641 (2007)] (here ovr(K) denotes the outer volume ratio of K), is not always sharp. Namely, we construct an example showing that there exists a symmetric convex body K which simultaneously has large outer volume ratio and large vertex index. Finally, we improve the constant in the latter bound for the case of the Euclidean ball from to , providing a completely new approach to the problem.

Original languageEnglish
Title of host publicationGeometric aspects of functional analysis
Subtitle of host publicationIsrael Seminar 2006–2010
EditorsBo'az Klartag, Shahar Mendelson, Vitali D. Milman
PublisherSpringer Heidelberg
Pages255-265
Number of pages11
ISBN (Electronic)978-3-642-29849-3
ISBN (Print)978-3-642-29848-6
DOIs
StatePublished - 2012

Publication series

NameLecture Notes in Math.
PublisherSpringer, Heidelberg
Volume2050
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

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