An $O(1)$ Bound for the KLS Constant

Abstract

The Kannan--Lov\'asz--Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $\psi_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We also prove that $C_P(\mu)\le C'$ for every isotropic log-concave probability measure $\mu$ on $\mathbb R^n$, with a universal constant $C'>0$.