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The Kannan-Lovász-Simonovits conjecture

Let K⊂RnK \subset \mathbb{R}^n be a convex body in isotropic position (barycenter at the origin, identity covariance). Kannan, Lovász and Simonovits conjectured that the isoperimetric inequality on KK is saturated, up to a universal constant, by half-spaces: Voln−1(∂A∩K)≥cmin⁡{Voln(A∩K),Voln(K∖A)}\mathrm{Vol}_{n-1}(\partial A \cap K) \ge c \min\{\mathrm{Vol}_n(A \cap K), \mathrm{Vol}_n(K \setminus A)\} with c>0c > 0 independent of nn and KK. Equivalent forms: exponential concentration of 1-Lipschitz functions with a dimension-free rate, or a Poincaré inequality with a universal constant, for every isotropic log-concave measure. Is the KLS constant bounded independently of the dimension?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI co-developed
Method
Argument
Field
Asymptotic convex geometry; high-dimensional probability
Posed by
Ravi Kannan, László Lovász and Miklós Simonovits (Discrete & Computational Geometry, 1995)
Year posed
1995
Years open
31y
Solved
2026-10-04
Model
ChatGPT (version not stated)
Vendor
OpenAI
Collaborators
Pierre Bizeul, Boaz Klartag, Joseph Lehec
Verification
Unreviewed
Publication
Preprint
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the conjecture in full, for all isotropic log-concave measures: CP(μ)≤CC_P(\mu) \le C with CC universal. The route bounds the third cumulant, propagates the estimate to cumulants of every order by Eldan's stochastic localization, realizes derivatives of tilt averages as mixed cumulants through a suspension construction, and closes with the Song-Zhang criterion. The previous best bounds were Klartag's Clog⁡nC\sqrt{\log n} and Song and Zhang's C⋅16log⁡∗nC \cdot 16^{\log^* n} (1 October 2026). Priority: Song and Zhang revised their preprint on 4 October 2026, the day this paper was posted, to claim the same O(1)O(1) bound by their own route; the two are parallel claims of the same result. Consequences include the thin-shell and variance conjectures, already known to follow from KLS.

What the AI did

From the paper's acknowledgements: "Most proofs and mathematical ideas in this paper were found by ChatGPT; a notable exception is the idea to use suspension which was suggested by the authors. The role of the authors has been mostly to understand these proofs and improve their exposition." The suspension construction is the step that turns tilt-average derivatives into cumulants of a larger isotropic log-concave measure, so a named human idea carries part of the argument. The model version is not stated.

Verification

No independent mathematician has checked this yet. Read here on 7 October 2026 from the arXiv source: Theorem 1.1 states a universal bound on the Poincaré constant of every isotropic log-concave measure in every dimension, which is the KLS conjecture in its Poincaré form and contains the original convex-body statement. The proof was not refereed here. The authors are leading researchers of the area, not independent reviewers, and there is no formal proof. The argument's last step rests on Song and Zhang's criterion (arXiv:2610.01447, Theorem 5.1), which the paper re-proves in its Section 3.

Sources

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