Geometric Analysis and Topology Seminar

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On the Connection Between the Spectral Gap of Convex Bodies and the Variance Conjecture

Speaker: Ronan Eldan, Microsoft Research

Location: Warren Weaver Hall 101

Date: Friday, November 22, 2013, 12:30 p.m.

Synopsis:

We consider the uniform measure over a high-dimensional isotropic convex body. We prove that, up to logarithmic factors, the isoperimetric minimizers are ellipsoids. Equivalently, we show that up to a logarithmic factor, the "worst-behaving" functions in the corresponding poincare inequality are quadratic functions. We thus establish a connection between two well-known conjectures regarding the uniform measure over a high dimensional convex body, namely the Thin-Shell conjecture and the conjecture by Kannan-Lovasz-Simonovits (KLS), showing that a positive answer to the former will imply a positive answer to the latter (up to a logarithmic factor). Our proof relies on the analysis of the eigenvalues of a certain random-matrix-valued stochastic process related to a convex body.