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.

References

Conjecture [Kannan-Lovász-Simonovits [KLS]] For any log-concave probability measure μ on ℝn, ‖Cov(μ)‖{op} ≤ C_P(μ) ≤ C * ‖Cov(μ)‖{op} where C > 0 is a universal constant.

— Isoperimetric inequalities in high-dimensional convex sets  (2406.01324 - Klartag et al., 2024) in Conjecture [Kannan–Lovász–Simonovits], Section 1 (The Poincaré inequality)

For general log-concave distributions, the Kannan--Lovász--Simonovits (KLS) conjecture asserts that the Poincaré constant is at most a universal multiple of the largest eigenvalue of the covariance. In isotropic position, this amounts to the bound $C_{\mathrm P}(P)\lesssim1$; the Kothari--Steinhardt argument would then yield \Cref{thm:main}.

— On the SoS Certifiability of Log-Concave Distributions  (2609.30105 - Storozhenko, 24 Sep 2026) in Section 1, subsection “Poincaré-Based Certificates and KLS”

Having its origin in theoretical computer science, the Kannan--Lovász--Simonovits conjecture (see, e.g., [AGB2015]) is arguably the most famous open problem in asymptotic geometric analysis and high-dimensional probability theory today; the currently best known result is due to B. Klartag [K2023]. Even though the bound has been improved considerably in recent years, it is not clear whether the conjecture is indeed true.

— The large and moderate deviations approach in geometric functional analysis  (2403.03940 - Prochno, 2024) in Subsection “Large deviations, moderate deviations, and the KLS conjecture”

The resulting bound is nevertheless unbounded in n and therefore does not resolve the KLS conjecture.

— An $O(1)$ Bound for the KLS Constant  (2610.01447 - Song et al., 1 Oct 2026) in Section 1, Introduction, subsection “Our results”