Kannan–Lovász–Simonovits (KLS) Conjecture

Prove that there exists a universal constant C > 0 such that for every log-concave probability measure μ on ℝ^n, the Poincaré constant C_P(μ) is within a universal multiplicative factor of the operator norm of its covariance matrix: ‖Cov(μ)‖_op ≤ C_P(μ) ≤ C · ‖Cov(μ)‖_op.

Background

The KLS conjecture predicts a dimension-free characterization of the Poincaré (spectral gap) constant for all log-concave measures. It asserts that linear functions essentially optimize the Poincaré inequality for this class, up to a universal constant.

This conjecture underlies major advances in convex geometry, probability, and algorithms, and is equivalent to a Cheeger-type isoperimetric formulation. It is consistent with known exact results for Gaussian and product measures and with optimal scaling on classical convex bodies.