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Bizeul's conjecture: a dimension-free log-Sobolev inequality for log-concave measures with subgaussian linear marginals

Call a centered random vector XX linearly aa-subgaussian if sup⁡∣θ∣=1Eexp⁡(⟨X,θ⟩2/a2)≤2\sup_{|\theta|=1}\mathbb E\exp(\langle X,\theta\rangle^2/a^2)\le2. A log-Sobolev inequality forces such tails, and Bakry-Emery gives one under uniform convexity, but log-concavity alone does not. Bobkov's criterion gives a constant of order na2na^2 and Bizeul improved this to order n a2\sqrt n\,a^2, with the dimension-free bound for rotationally invariant laws. Bizeul conjectured that subgaussian linear marginals suffice; Klartag and Lehec list it as Conjecture 76. Is there a universal CC such that every centered log-concave probability measure μ\mu on Rn\mathbb R^n that is linearly aa-subgaussian satisfies Entμ(f2)≤Ca2∫∣∇f∣2dμ\mathrm{Ent}_\mu(f^2)\le Ca^2\int|\nabla f|^2d\mu?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Asymptotic convex geometry; functional inequalities for log-concave measures
Posed by
Pierre Bizeul (On the log-Sobolev constant of log-concave vectors, formulated 2023, J. Funct. Anal. 290 (2026), Conjecture 2); Klartag and Lehec, Bull. AMS (2025), Conjecture 76
Year posed
2023
Years open
3y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is an absolute CC such that for every nn, every centered log-concave probability μ\mu on Rn\mathbb R^n with a density and linear subgaussian parameter aa, Entμ(f2)≤Ca2∫∣Df∣2dμ\mathrm{Ent}_\mu(f^2)\le Ca^2\int|Df|^2d\mu for all f∈Cc∞f\in C_c^\infty; this includes uniform measures on convex bodies and needs no smoothness or curvature. By Otto-Villani it gives the transport-entropy inequality W2(ν,μ)2≤Ca2H(ν∣μ)W_2(\nu,\mu)^2\le Ca^2H(\nu|\mu) with the same constant. Not given: a value of CC, or anything for non-log-concave measures.

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Bizeul's Conjecture 2 as the paper states it; proof not refereed. No Lean formalization exists for this family. The paper states its scope: centered log-concave laws with a Lebesgue density (degenerate laws on lower-dimensional subspaces are reduced to this), the specific ψ2\psi_2 marginal normalization above, and an unspecified absolute constant CC.

Source

Changelog1 change

Discussion