# Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures > Log-concave laws with bell-curve tails in every direction obey a dimension-free log-Sobolev bound, the paper claims. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-093/ Last updated: 2026-10-14 **Log-concave laws with bell-curve tails in every direction obey a dimension-free log-Sobolev bound, the paper claims.** **Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 23, 2026 · **Field:** Convex and metric geometry · **Entry:** 093 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. A logarithmic Sobolev inequality bounds how uneven a function can be by how steep it is on average. The question is whether bell-curve tails in every straight direction are enough to get one, for one-hump (log-concave) distributions. This paper claims they are, with constant C times a squared, where C is the same in every dimension. Earlier bounds grew with the dimension, like n or the square root of n. No Lean formalization is listed. ## At a glance - **First step:** 2023 (Pierre Bizeul, who formulated the conjecture) - **Significance:** Major. If the proof holds, it settles a stated conjecture on log-concave laws by removing all dependence on dimension. - **Trust:** Claimed, not checked - **Field:** Convex and metric geometry (1 paper) ## The explanation Start with the bell curve, the standard normal distribution. Draw X from it and compute the average of exp(X²/a²). The answer is 1 divided by the square root of 1 − 2/a². That equals 2 exactly when a² = 8/3, so a is about 1.63. In the paper's terms, X is a-subgaussian with a ≈ 1.63. Being linearly a-subgaussian means every straight projection passes this test with the same a[1]. Now take a standard bell curve in n dimensions. Every straight projection of it is again a standard bell curve. So the same a ≈ 1.63 works in every dimension, even a million. The question is whether that kind of tail control buys something much stronger. The stronger thing is a logarithmic Sobolev inequality. It bounds the entropy of a function [a number measuring how uneven its values are] by its Dirichlet energy [the average squared steepness][2]. It also gives bell-curve concentration for every Lipschitz function [a function that cannot change faster than a fixed rate][3]. Write the bound as entropy ≤ constant × energy. For the bell curve, a classical result of Bakry and Émery keeps the constant fixed in every dimension[4]. Their method needs the density to curve downward at a guaranteed rate. This paper works with log-concave laws [densities whose logarithm is concave, giving one hump and light tails]. Log-concavity alone promises no guaranteed downward curvature[5]. So the classical method has nothing to push against. The question is whether bell-curve tails in every straight direction can replace that curvature[6]. The tails are necessary, up to universal constants[7]. Pierre Bizeul formulated their sufficiency as a conjecture in 2023[8]. Earlier bounds grew with dimension. A criterion of Bobkov from 1999 gives a constant of order n × a²[9]. Bizeul improved this to order √n × a²[10]. In 10,000 dimensions, those are 10,000a² and 100a². Bizeul also proved the dimension-free bound for laws that look the same in every direction[11]. The paper claims the constant is C × a², with one universal C[12]. The same C works in every dimension[13]. No smoothness of the density or positive curvature is needed[14]. So the theorem includes uniform measures on convex bodies [solid shapes with no dents][15]. The theorem names no numerical value for C. It needs the law to be centered, with average zero, and to have a density. The hard part is the gap between hypothesis and conclusion. The tail condition speaks only about straight, linear functions. The conclusion controls the entropy of every nonlinear test function[16]. The paper argues by contradiction. A failure would give laws whose log-Sobolev constant stays at 1 while their tail scale a shrinks to zero[17]. The paper then derives two incompatible descriptions of how these laws fluctuate[18]. There is also a standard consequence. By a 2000 result of Otto and Villani, the bound implies a transport–entropy inequality with the same constant[19]. That inequality limits how far a distribution can be moved, in an average-distance sense, by its relative entropy. How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published. **What it doesn't show.** This does not give a numerical value for the universal constant C, and it does not cover laws without the log-concave shape. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major only if the proof holds. The constant C is universal but not given as a number, and the law must be centered and log-concave. No Lean formalization is listed and no specialist review is published. ## Words used - **Logarithmic Sobolev inequality:** A bound on the entropy of a function by a constant times its average squared steepness. - **Entropy:** Here, a number that measures how uneven the values of a function are under a given distribution. - **Log-concave:** A density whose logarithm is concave, so it has a single hump and light tails. - **Linearly subgaussian:** Every straight projection has tails no heavier than a bell curve of a fixed width a. - **Dimension-free:** The constant stays the same no matter how many dimensions the space has. - **Lipschitz function:** A function that cannot change faster than a fixed rate. - **Convex body:** A solid shape with no dents: the segment between any two of its points stays inside. ## The original paper - [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf) - September 23, 2026 ## Quoted lines from the paper (19) [1] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > This definition concerns only linear functions of \(X\). [2] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > A logarithmic Sobolev inequality controls the entropy of a function by its Dirichlet energy. [3] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Besides implying a Poincar\'e inequality, it gives Gaussian concentration for Lipschitz functions. [4] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The diffusion approach of Bakry and \'Emery gives the inequality for a density $e^{-V}$ when $D^2V\ge\kappa\Id$ with $\kappa>0$: in the convention used below, its constant is at most $\kappa^{-1}$. [5] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Log-concavity alone asserts only $D^2V\ge0$ when the potential is smooth. [6] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The question addressed here is whether a global assumption on the tails of linear functions can supply the logarithmic Sobolev bound without positive curvature. [7] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Uniformly subgaussian linear marginals are necessary, up to universal constants, by the same Herbst argument. [8] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Bizeul formulated their sufficiency within the log-concave class in 2023. [9] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Under (…), H\"older's inequality gives $\E\exp(|X|^2/(na^2))\le2$, so that criterion yields a coefficient of order $na^2$. [10] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Thus for linear subgaussian scale $a$ it gives an entropy coefficient of order $\sqrt n\,a^2$. [11] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Bizeul also obtained the dimension-free conclusion for rotationally invariant log-concave laws. [12] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > We prove that every centered log-concave probability measure with a Lebesgue density on $\mathbb R^n$ and linear subgaussian parameter $a$ satisfies $\operatorname{Ent}_\mu(f^2)\le Ca^2\int|Df|^2\,d\mu$ for compactly supported smooth $f$, with one universal constant $C$. [13] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The same constant works in every dimension. [14] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > No smoothness of the density or positive lower curvature bound is required. [15] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > In particular, the theorem includes uniform measures on convex bodies. [16] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The theorem's marginal hypothesis concerns more than covariance: it controls the tails of every linear function, while its conclusion controls the entropy of nonlinear tests. [17] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > Failure of Theorem … would give, after truncation, a weak quadratic tilt, and dilation, a sequence of compactly supported log-concave laws \(\mu\) such that \(R(\mu)=1\) while their linear subgaussian parameters tend to zero. [18] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The argument derives incompatible descriptions of fluctuations associated with these normalized laws. [19] From [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf): > The Otto--Villani implication from logarithmic Sobolev to quadratic transport--entropy inequalities gives the following standard consequence, with the same constant. ## Other sources - [L. Gross, “Logarithmic Sobolev inequalities”, American Journal of Mathematics 97(4) (1975), 1061–1083](https://www.jstor.org/stable/2373688) - [D. Bakry and M. Émery, “Diffusions hypercontractives”, Séminaire de probabilités XIX, Lecture Notes in Math. 1123 (1985), 177–206](https://numdam.org/item/SPS_1985__19__177_0/) - [S. G. Bobkov, “Isoperimetric and analytic inequalities for log-concave probability measures”, Annals of Probability 27(4) (1999), 1903–1921](https://doi.org/10.1214/aop/1022874820) - [P. Bizeul, “On the log-Sobolev constant of log-concave vectors”, Journal of Functional Analysis 290(9) (2026), 111368; arXiv:2306.12997](https://arxiv.org/abs/2306.12997v3) - [F. Otto and C. Villani, “Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality”, Journal of Functional Analysis 173(2) (2000), 361–400](https://doi.org/10.1006/jfan.1999.3557) - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [The optimal order of convex-body covering density](https://closedform.nishachay.tech/e/openai-math-2026-092/) - Convex and metric geometry - [Subpolynomial dimension reduction in L p](https://closedform.nishachay.tech/e/openai-math-2026-094/) - Convex and metric geometry - [Logarithmic and L p Brunn–Minkowski inequalities and the B-conjecture](https://closedform.nishachay.tech/e/openai-math-2026-091/) - Convex and metric geometry - [Hyperbolicity cones without semidefinite lifts](https://closedform.nishachay.tech/e/openai-math-2026-095/) - Convex and metric geometry - [Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy](https://closedform.nishachay.tech/e/openai-math-2026-090/) - Convex and metric geometry - [Convex and metric geometry](https://closedform.nishachay.tech/science/mathematics/convex-and-metric-geometry/) - Field · 15 results - [Open since the 2020s](https://closedform.nishachay.tech/open-since/2020s/) - Era · 9 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20093%5D%20)