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

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 ${\psi }_n\lesssim {\log }^{1\mathrm{/}4}n$, due to Letwin [Let26]. We prove that ${\psi }_n\le C\cdot 4^{{\log }^{\ast }(n+2)}$ for a universal constant $C$, where ${\log }^{\ast }x$ is the least number of successive natural logarithms needed to bring $x$ to at most one. We also prove that $C_P(\mu )\le C^{\mathrm{\prime }}{16}^{{\log }^{\ast }(n+2)}$ for every isotropic log-concave probability measure $\mu$ on ${\mathbb{R}}^n$, with a universal constant $C^{\mathrm{\prime }}>0$.