Kannan–Lovász–Simonovits (KLS) conjecture for log-concave measures

Establish the Kannan–Lovász–Simonovits conjecture in the form relevant to this work: For any log-concave probability measure μ on ℝ^d normalized so that the Lipschitz Poincaré constant satisfies C_P^{Lip}(μ)=1 (equivalently, Ψ_μ(0)=1), determine whether the largest variance over all directions is uniformly bounded below by a universal constant independent of d, namely prove c ≤ ∥Cov(μ)∥_{op} ≤ 1 for some universal c>0. Equivalently, prove that the Poincaré inequality for log-concave measures is saturated by linear functions up to universal constants, so that C_P(μ) and C_P^{Lip}(μ) are within universal multiplicative factors of ∥Cov(μ)∥_{op}.

Background

The paper studies regression of 1-Lipschitz functions under log-concave measures and normalizes risk by assuming the Lipschitz Poincaré constant Ψ_μ(0)=1. Under this normalization, the strongest dimension-free control on concentration would follow from the Kannan–Lovász–Simonovits (KLS) conjecture, which predicts that the Poincaré inequality for log-concave measures is actually saturated by linear functions.

Concretely, KLS would imply that normalizing the Lipschitz Poincaré constant (Ψμ(0)=1) is equivalent to normalizing the largest directional variance by a universal constant c>0, i.e., c ≤ ∥Cov(μ)∥{op} ≤ 1. The current best known bounds give c≈1/log n, highlighting a gap that the conjecture seeks to close with a dimension-free constant.