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.

1 paper September 23, 2026 Explained by AI

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.

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 aSource 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]Source 2. It also gives bell-curve concentration for every Lipschitz function [a function that cannot change faster than a fixed rate]Source 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 dimensionSource 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 curvatureSource 5. So the classical method has nothing to push against. The question is whether bell-curve tails in every straight direction can replace that curvatureSource 6. The tails are necessary, up to universal constantsSource 7. Pierre Bizeul formulated their sufficiency as a conjecture in 2023Source 8.

Earlier bounds grew with dimension. A criterion of Bobkov from 1999 gives a constant of order n × a²Source 9. Bizeul improved this to order √n × a²Source 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 directionSource 11.

The paper claims the constant is C × a², with one universal CSource 12. The same C works in every dimensionSource 13. No smoothness of the density or positive curvature is neededSource 14. So the theorem includes uniform measures on convex bodies [solid shapes with no dents]Source 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 functionSource 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 zeroSource 17. The paper then derives two incompatible descriptions of how these laws fluctuateSource 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 constantSource 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 (PDF, opens in a new tab)
September 23, 2026

All quoted lines (19)

This definition concerns only linear functions of $X$ .
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
A logarithmic Sobolev inequality controls the entropy of a function by its Dirichlet energy.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Besides implying a Poincar\'e inequality, it gives Gaussian concentration for Lipschitz functions.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
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 $\kappa^{-1}$ .
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Log-concavity alone asserts only $D^2V\ge0$ when the potential is smooth.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
The question addressed here is whether a global assumption on the tails of linear functions can supply the logarithmic Sobolev bound without positive curvature.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Uniformly subgaussian linear marginals are necessary, up to universal constants, by the same Herbst argument.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Bizeul formulated their sufficiency within the log-concave class in 2023.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Under (…), H\"older's inequality gives $\E\exp(|X|^2/(na^2))\le2$ , so that criterion yields a coefficient of order $na^2$ .
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Thus for linear subgaussian scale $a$ it gives an entropy coefficient of order $\sqrt n\,a^2$ .
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
Bizeul also obtained the dimension-free conclusion for rotationally invariant log-concave laws.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
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 $C$ .
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
The same constant works in every dimension.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
No smoothness of the density or positive lower curvature bound is required.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
In particular, the theorem includes uniform measures on convex bodies.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
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.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
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.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
The argument derives incompatible descriptions of fluctuations associated with these normalized laws.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text
The Otto–Villani implication from logarithmic Sobolev to quadratic transport–entropy inequalities gives the following standard consequence, with the same constant.
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures Back to text

Other sources

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