KLS conjecture finally solved: Zhao Song and Xinzhi Zhang post 140-page O(1) bound proof
basedjensen · x · 2026-10-06
The Kannan–Lovász–Simonovits (KLS) conjecture has finally been resolved. Zhao Song and Xinzhi Zhang posted a 140-page arXiv paper, An O(1) Bound for the KLS Constant, proving that ψₙ ≤ C for a universal constant C, i.e., every isotropic log-concave probability measure on Rⁿ has a Cheeger constant bounded below by a universal positive constant. They also prove a universal bound CP(μ) ≤ C' on the Poincaré constant.
The KLS conjecture is a central open problem in convex geometry and probability, closely tied to isoperimetry in high-dimensional distributions and the complexity analysis of sampling algorithms. Congratulations were extended to the authors, one of whom is at Microsoft Research.
Related event: Two AI-assisted papers claim to resolve the KLS conjecture(7 posts)→
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