Kannan–Lovász–Simonovits (KLS) Conjecture

Prove the Kannan–Lovász–Simonovits conjecture: Establish the existence of a universal constant C > 0 such that for every dimension d ∈ N, every centered random vector X in R^d with a log-concave distribution, and every locally Lipschitz function f: R^d → R with finite variance, the Poincaré-type inequality Var[f(X)] ≤ C · λ_X^2 · E[||∇f(X)||_2^2] holds, where λ_X^2 := sup_{θ ∈ S^{d−1}} E[⟨X, θ⟩^2].

Background

The Kannan–Lovász–Simonovits (KLS) conjecture is a central problem at the interface of asymptotic geometric analysis, high-dimensional probability, and theoretical computer science. It predicts a dimension-free Poincaré inequality for all isotropic log-concave measures, implying strong isoperimetric and mixing properties fundamental to efficient sampling and volume computation algorithms for high-dimensional convex bodies.

The paper highlights recent partial progress, including improved bounds culminating in Klartag’s 2023 result achieving a √(log d) factor, yet the conjecture remains unresolved. It further connects the conjecture to the paper of large and moderate deviations, suggesting that constructing certain deviation principles for isotropic log-concave random vectors could provide a route to disprove the conjecture.

Within this survey, the authors formulate the conjecture precisely via a variance bound controlled by the direction-wise second moment λ_X^2 and the expected squared gradient norm, underscoring both its importance and its unresolved status.