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KLS Conjecture for Quadratic Forms

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?

Result
Proved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Argument
Field
Asymptotic convex geometry
Posed by
Ravi Kannan, László Lovász & Miklós Simonovits
Year posed
1995
Years open
31y
Solved
2026-07-27
Model
ChatGPT-5.6 Pro
Vendor
OpenAI
Collaborators
Brayden Letwin
Verification
Unreviewed
Publication
Preprint
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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

What the AI did

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

Verification

Author-checked arXiv preprint proving the quadratic-form case with constant 2 and deriving the global estimate ψn≤Clog⁡1/4n\psi_n \le C \log^{1/4} n. Not yet peer-reviewed.

Source

arXiv:2607.24164 - The KLS constant is O(log^(1/4) n)

Discussion