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KLS Conjecture for Quadratic Forms

Geometry & topology · posed by Ravi Kannan, László Lovász & Miklós Simonovits, 1995 · solved

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Statement

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is Var⁡⟨MX,X⟩≤C E∣∇⟨MX,X⟩∣2\operatorname{Var}\langle MX, X\rangle \le C\, \mathbb{E}|\nabla\langle MX, X\rangle|^2 for every symmetric MM?

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Brayden Letwin, using ChatGPT-5.6 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The key argument was developed in collaboration with ChatGPT-5.6 Pro and checked by the author.

    with constant 2; also improves the global KLS bound to O(log⁡1/4n)O(\log^{1/4} n)

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