Eldan's Stochastic Localization and the KLS Hyperplane Conjecture: An Improved Lower Bound for Expansion

Abstract

We show that the KLS constant for n-dimensional isotropic logconcavemeasures is O(n^{1/4}), improving on the current best bound ofO(n^{1/3}√{\log n}). As corollaries we obtain the same improvedbound on the thin-shell estimate, Poincar\e constant and Lipschitzconcentration constant and an alternative proof of this bound forthe isotropic constant; it also follows that the ball walk for samplingfrom an isotropic logconcave density in \R^{n} converges in O^{*}(n^{2.5})steps from a warm start.