The logarithmic Sobolev conjecture for subgaussian log-concave measures
From OpenAI Aftermath
The logarithmic Sobolev conjecture for subgaussian log-concave measures asks whether a centered log-concave probability measure on Rn whose linear functions all have Gaussian tails, with a common subgaussian parameter a, satisfies a logarithmic Sobolev inequality with constant of order a2, independent of the dimension. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims this statement. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.
Background
- As recounted in the introduction of the manuscript:
- Gross's work connected the Gaussian logarithmic Sobolev inequality with hypercontractivity, and the Bakry–Émery criterion gives the inequality under a positive lower bound on the Hessian of the potential. Log-concavity alone gives only a nonnegative lower bound.
- A logarithmic Sobolev inequality implies Gaussian concentration for Lipschitz functions (Herbst), so uniformly subgaussian linear marginals are necessary up to constants. Under log-concavity, Gaussian concentration for all Lipschitz functions gives a logarithmic Sobolev inequality by results of E. Milman and Bobkov.
- Bizeul formulated the sufficiency of subgaussian linear marginals within the log-concave class in 2023, proved a bound with dimension dependence of order √n in the entropy coefficient, and proved the dimension-free statement for rotationally invariant laws. Klartag and Lehec discuss the conjecture in their survey (Conjecture 76). Bobkov's earlier norm-based criterion gives a coefficient of order na2.
The original result
- Claim. Theorem thm:main of A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures states that there is an absolute constant C such that for every n, every centered log-concave probability measure μ with a Lebesgue density on Rn whose linear marginals satisfy sup|θ|=1 E exp{⟨X,θ⟩2/a2} ≤ 2 satisfies Entμ(f2) ≤ Ca2 ∫ |∇f|2 dμ for every compactly supported smooth f. No smoothness or positive curvature of the density is assumed, so uniform measures on convex bodies are included.
- Consequence, as stated by the manuscript. By the Otto–Villani theorem, the same measures satisfy the quadratic transport–entropy inequality W2(ν,μ)2 ≤ Ca2 H(ν | μ) with the same constant.
- Method, as described: an argument by contradiction from a normalized sequence of counterexamples, using Gaussian prediction identities going back to Guo, Shamai and Verdú, an entropy-loss lemma based on repeated observations of independent copies, a hierarchy of symmetric tensors and a matrix square root whose diffusion law is lognormal.
- Sources. At commit
1581d4c: A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures. - Formalization. The repository's Lean documentation has no scope note for this family at that commit.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 23 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | mxym, Semiconvex Gaussian entropy loss (mxym/math, notes/semiconvex-gaussian-entropy) | Extension (refines a lemma) | Refines the manuscript's finite-observation entropy-loss lemma, Lemma 4.2, from log-concave references to references with Hess(−log q) ≥ −κI: the entropy lost under Gaussian convolution up to time z < 1/κ is bounded by the initial-score prediction energy with weight (1 − κr)−2. The note states that the constants are sharp, gives a finite counterexample to the unweighted comparison, and shows that no finite factor depending only on the curvature and time exists at z ≥ 1/κ. It attributes the proof architecture to the manuscript and records one correction to the domain of a helper lemma that does not affect the main theorem. | Repository note prepared with AI assistance; its README reports an independent analytic audit within the package and makes no novelty claim. |
| 2026-10-13 | M. Colbrook and contributors, AIM — Open Applied Problems (GitHub), problem 092 | Catalogue (partial) | In its 13 October status review of problem 092, the dimension-free Kannan–Lovász–Simonovits inequality, the collection recorded the manuscript as a partial result: linearizing the logarithmic Sobolev inequality gives the Poincaré bound for isotropic log-concave measures whose linear subgaussian parameter is bounded by an absolute constant, while general isotropic log-concave measures need not be subgaussian. Since 15 October the problem is marked "solution claimed" on the basis of a separate work, Bizeul, Klartag and Lehec's arXiv 2610.05474. | The status reviews are described as AI statement-level comparisons, not independent audits of complete proofs. The collection is not affiliated with the American Institute of Mathematics. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_422 of the manuscript in JJardine919/openai-math-audit | Review (AI-assisted audit) | Reports no gap after reading all seven sections, with hand and numerical checks of the main steps, including numerical tests of the entropy-loss lemma on Gaussian and one-dimensional log-concave examples. It states that the operator-tail lemma, the block rounding and input clipping, and the choice of limit orders were only lightly checked, so that a subtle gap there cannot be ruled out. | Part of an audit of the collection's manuscripts that the audit describes as complete. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
