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arXiv · 2610.07728

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

Krishnakumar Balasubramanian·Shiva Kasiviswanathan
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Abstract

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.

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Krishnakumar Balasubramanian, Shiva Kasiviswanathan. 2026-10-06. A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728

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