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Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates

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Date
2014
Publisher city
Paris
Notes
Lecture Notes in Mathematics n°2116
Link to item file
https://arxiv.org/abs/1306.3696v2
Dewey
Probabilités et mathématiques appliquées
Sujet
Gaussian vector; Thin-Shell Estimates; Chaining techniques
DOI
http://dx.doi.org/10.1007/978-3-319-09477-9_9
Book title
Geometric Aspects of Functional Analysis. Israel Seminar (GAFA) 2011-2013
Author
Bo'az Klartag, Emanuel Milman
Publisher
Springer
Publisher city
Berlin Heidelberg
Year
2014
ISBN
978-3-319-09476-2
Book URL
10.1007/978-3-319-09477-9
URI
https://basepub.dauphine.fr/handle/123456789/14512
Collections
  • CEREMADE : Publications
Metadata
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Author
Eldan, Ronen
Lehec, Joseph
Type
Chapitre d'ouvrage
Item number of pages
107-122
Abstract (EN)
Chaining techniques show that if X is an isotropic log-concave random vector in R n and Γ is a standard Gaussian vector then EX ≤ Cn 1/4 EΓ for any norm · , where C is a universal constant. Using a completely different argument we establish a similar inequality relying on the thin-shell constant σn = sup Var(|X|); X isotropic and log-concave on R n . In particular, we show that if the thin-shell conjecture σn = O(1) holds, then n 1/4 can be replaced by log(n) in the inequality. As a consequence, we obtain certain bounds for the mean-width, the dual mean-width and the isotropic constant of an isotropic convex body. In particular, we give an alternative proof of the fact that a positive answer to the thin-shell conjecture implies a positive answer to the slicing problem, up to a logarithmic factor.

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