An $O(4^{\log^* n})$ Bound for the KLS Constant

Source abstract

The Kannan--Lovász--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. The best previous upper bound is $ψ_n\lesssim\log^{1/4}n$, due to Letwin [Let26]. We prove that $ψ_n\le C \cdot 4^{\log^*(n+2)}$ for a universal constant $C$, where $\log^*x$ is the least number of successive natural logarithms needed to bring $x$ to at most one. We also prove that $C_P(μ)\le C'16^{\log^*(n+2)}$ for every isotropic log-concave probability measure $μ$ on $\mathbb R^n$, with a universal constant $C'>0$.