An O(4log∗n) Bound for the KLS Constant

The Kannan–Lovász–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on Rn has a Cheeger constant bounded below by a universal positive constant. The best previous upper bound is ψn≲log1/4n, due to Letwin [Let26]. We prove that ψn≤C⋅4log∗(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 CP(μ)≤C′16log∗(n+2) for every isotropic log-concave probability measure μ on Rn, with a universal constant C′>0.