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

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Abstract

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.

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  1. School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Ronen Eldan

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  1. Ronen Eldan
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Correspondence to Ronen Eldan.

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Supported in part by the Israel Science Foundation and by a Marie Curie Grant from the Commission of the European Communities.

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Eldan, R. Thin Shell Implies Spectral Gap Up to Polylog via a Stochastic Localization Scheme. Geom. Funct. Anal. 23, 532–569 (2013). https://doi.org/10.1007/s00039-013-0214-y

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