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

TL;DR

This paper links the thin shell conjecture and Kannan-Lovasz-Simonovits conjecture in high-dimensional convex geometry, showing their bounds are equivalent up to logarithmic factors using a stochastic localization scheme.

Contribution

It establishes the equivalence of two major open problems in convex geometry up to logarithmic factors and introduces a stochastic localization scheme for log-concave measures.

Findings

Optimal surface area to volume ratio is attained on ellipsoids up to logarithmic factors.

A positive resolution of the thin shell conjecture would yield optimal dimension dependence in a form of the Brunn-Minkowski inequality.

The paper develops a stochastic localization scheme for log-concave measures.

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, Lovasz, 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.