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

By Ronen Eldan. 2012

160 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b4f5bbdea992bd425d36d23f68a0a4cc4d2a1bcc

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

A two-sided estimate for the Gaussian noise stability deficit (2013, methodology): “Equations of a similar nature, where the initial density is an arbitrary function seem to be rather useful tool for proving concentration inequalities, as demonstrated in [10–12].”
Bypassing KLS: Gaussian Cooling and an O^*(n3) Volume Algorithm (2014, background): “Indeed, it captures two well-known and much older conjectures from convex geometry, the slicing (or hyperplane) conjecture and the thin-shell conjecture (these were all shown to be equivalent in a certain sense recently [10, 11]), and thus has effectively evaded resolution for nearly a half-century.”