A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures

The Kannan–Lovász–Simonovits (KLS) conjecture asks whether isotropic log-concave probability measures satisfy a Poincaré inequality with a constant independent of dimension. In this paper, we establish a dimension-free Poincaré inequality and hence a universal positive lower bound for the Euclidean Cheeger constant, thereby proving the KLS conjecture. Our proof has three steps. First, we study integration operators that undo differentiation and prove a curvature estimate that is uniform in the number of tensor indices. Second, an operator lemma turns bounds for low-degree polynomials into bounds for any number of integrations, with one common multiplicative factor. Third, we use stochastic localization to transfer the integration bounds back to polynomial norms.