Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates
Eldan, Ronen; Lehec, Joseph (2014), Bounding the Norm of a Log-Concave Vector Via Thin-Shell Estimates, in Bo'az Klartag, Emanuel Milman, Geometric Aspects of Functional Analysis. Israel Seminar (GAFA) 2011-2013, Springer : Berlin Heidelberg, p. 107-122. 10.1007/978-3-319-09477-9_9
External document linkhttps://arxiv.org/abs/1306.3696v2
Book titleGeometric Aspects of Functional Analysis. Israel Seminar (GAFA) 2011-2013
Book authorBo'az Klartag, Emanuel Milman
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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.
Subjects / KeywordsGaussian vector; Thin-Shell Estimates; Chaining techniques
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