Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and the conjecture by Kannan, Lovász, and Simonovits , showing...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 23; no. 2; pp. 532 - 569
Main Author Eldan, Ronen
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.04.2013
Subjects
Analysis
Mathematics
Mathematics and Statistics
Convex Body
Isoperimetric Inequality
Thin Shell
Logarithmic Sobolev Inequality
Universal Constant
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Summary:We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and the conjecture by Kannan, Lovász, and Simonovits , showing that the corresponding optimal bounds are equivalent up to logarithmic factors. In particular we prove that, up to logarithmic factors, the minimal possible ratio between surface area and volume is attained on ellipsoids. We also show that a positive answer to the thin shell conjecture would imply an optimal dependence on the dimension in a certain formulation of the Brunn–Minkowski inequality. Our results rely on the construction of a stochastic localization scheme for log-concave measures.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-013-0214-y