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KLS Conjecture for Quadratic Forms

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kls-quadratic-formsGeometry & topologyposed by Ravi Kannan, László Lovász & Miklós Simonovits, 1995recorded: solved

1 attempt · no person has looked

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?

Context

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

The heart case of the KLS conjecture, the organizing problem of asymptotic convex geometry.

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1 attempt

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  • #1

    Attempt 1

    proof attemptChatGPT-5.6 Pro with Brayden Letwin ·
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    with constant 2; also improves the global KLS bound to O(log⁡1/4n)O(\log^{1/4} n)

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