An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture

TL;DR

This paper establishes an almost constant lower bound for the isoperimetric coefficient in the KLS conjecture, improving the dimension dependency and impacting various conjectures and algorithms related to log-concave measures.

Contribution

It provides a tighter lower bound on the isoperimetric coefficient in the KLS conjecture with dimension dependency $d^{-o_d(1)}$, surpassing previous bounds.

Findings

Lower bound has dimension dependency $d^{-o_d(1)}$

Tighter than previous $d^{-1/4}$ bound for large dimensions

Implications for Bourgain's slicing, thin-shell conjecture, and MCMC algorithms

Abstract

We prove an almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. The lower bound has the dimension dependency $d^{-o_d(1)}$. When the dimension is large enough, our lower bound is tighter than the previous best bound which has the dimension dependency $d^{-1/4}$. Improving the current best lower bound of the isoperimetric coefficient in the KLS conjecture has many implications, including improvements of the current best bounds in Bourgain's slicing conjecture and in the thin-shell conjecture, better concentration inequalities for Lipschitz functions of log-concave measures and better mixing time bounds for MCMC sampling algorithms on log-concave measures.