An O(1) Bound for the KLS Constant

The Kannan–Lov´asz–Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on ${ℝ}^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that ${\psi }_n$ ≤ $C$ for a universal constant $C$ > 0, resolving the KLS conjecture. We also prove that $C_P(\mu )$ ≤ $C$′ for every isotropic log-concave probability measure $\mu$ on ${ℝ}^n$, with a universal constant $C$‘ > 0.