The KLS constant is O(log^1/4 n)

2026/07/27 by Brayden Letwin
#math.PR #math.FA #math.MG

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Abstract

We confirm the Kannan--Lovász--Simonovits conjecture for quadratic forms: if X ∼ μ is an isotropic log-concave random vector in ℝⁿ, then for any symmetric matrix M one has Var_{X ∼ μ}(⟨ MX,X⟩) ≤ 2 𝔼_{X ∼ μ}|∇⟨ MX,X⟩|². As an application, we apply the above to M=𝔼_{X ∼ μ}(⟨ X,θ⟩ X⊗ X) for θ∈ S^n-1 and show that the Kannan--Lovász--Simonovits constant ψ_n satisfies ψ_n≤ Clog^1/4n for some absolute constant C>0.

Citations

• Thin-shell bounds via parallel coupling
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• Stein's method, logarithmic Sobolev and transport inequalities
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• Poincare Inequalities and Moment Maps
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Related

• The Kannan-Lovász-Simonovits Conjecture
• Large deviations, moderate deviations, and the KLS conjecture
• On the Smallest Singular Value of Log-Concave Random Matrices
• Logarithmic bounds for isoperimetry and slices of convex sets
• Increasing Subsequences and Kronecker Coefficients