Thin shell implies spectral gap up to polylog via a stochastic localization scheme

Details Thin shell implies spectral gap up to polylog via a stochastic localization scheme Thin shell implies spectral gap up to polylog via a stochastic localization scheme

2012-03-05

arxiv.org/abs/1203.0893v1

Mathematics

Metric Geometry

25 pages, preliminary version

Scientific paper

We establish quantitative connections between two well-known open problems related to the uniform measure on a high dimensional convex body, namely, the thin shell conjecture, and the conjecture by Kannan, Lovasz, and Simonovits about the isoperimetric inequality for isotropic convex bodies. In particular we show that their corresponding optimal bounds are equivalent up to logarithmic factors. Moreover, we show that the thin shell conjecture will imply an optimal dependence on the dimension in a certain formulation of the Brunn-Minkowski inequality. Our resuts rely the construction of a stochastic localization scheme for log-concave measures.

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