Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

1 · The essentials

The manuscript studies how probability is spread across a space with any number of coordinates. It assumes that the average location is the origin and that the probability has a density whose logarithm bends downward. It also assumes that measurements in every straight-line direction have tails controlled on a common, Gaussian-like scale. Under these conditions, the manuscript shows an inequality linking how uneven the square of a smooth function is to how rapidly that function changes. The bound depends on the square of that common scale, but not on the number of coordinates. The abstract describes this as a positive resolution of the corresponding conjecture. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a rule that bounds the unevenness of a landscape by the steepness of its slopes. Here the notable feature is that the rule does not get worse when more coordinate directions are added, provided the probability distribution meets all the stated conditions.

Why it matters

A dimension-free bound avoids deterioration caused solely by increasing the number of coordinates. The abstract identifies precisely this bound, under log-concavity and uniformly subgaussian linear measurements, as the subject of a conjecture and claims its positive resolution.

2 · Going further

The precise setting is a centered log-concave probability measure $μ$ with a Lebesgue density on $R^{n}$ and linear subgaussian parameter $a$ . The main statement shown in the manuscript is that, for every compactly supported smooth function $f$ , $Ent_{μ} (f^{2}) \leq C a^{2} \int ∣ D f ∣^{2} d μ$ , where $C$ is one universal constant. Here $D f$ denotes the gradient, and the entropy on the left measures the unevenness of $f^{2}$ relative to its mean under $μ$ .

The key quantitative feature is that the same universal constant works independently of $n$ ; the distribution enters the stated bound through $a^{2}$ . The abstract describes this as resolving the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures. The supplied text does not specify the normalization used to define the linear subgaussian parameter.

The abstract states the result but gives no argument or proof strategy, so the main mechanism cannot be explained from this source alone. Only the main manuscript is supplied; no companion manuscripts or relationships to them are described.

Useful notions

Probability measures and densities
Concavity of a function
Gaussian-like tail bounds
Smooth functions and gradients
Entropy of a nonnegative function

What was known before / what is new

Before

The abstract identifies a dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures. The supplied source gives no account of earlier partial results, existing dimension-dependent bounds, or the conjecture's history.

New

The manuscript claims a logarithmic Sobolev bound with constant at most $C a^{2}$ for every measure satisfying the stated assumptions. The constant $C$ is universal, so it does not depend on the dimension. The abstract presents this as a positive resolution of the conjecture.

Limits and assumptions

The statement requires a centered, log-concave probability measure with a Lebesgue density on $R^{n}$ and a uniform linear subgaussian parameter $a$ . The stated inequality applies to compactly supported smooth functions; the supplied text does not state extensions beyond that class or beyond those measure assumptions. It gives neither an argument nor a precise normalization for $a$ , and it identifies no remaining open questions. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Centered measure: A probability measure whose mean is the origin.
Log-concave density: A density whose logarithm is concave wherever the density is positive.
Linear subgaussian parameter: A common scale controlling Gaussian-like tails of one-dimensional linear measurements of the distribution.
Dimension-free bound: A bound whose constant does not depend on the number of coordinates.
Logarithmic Sobolev inequality: An inequality bounding the entropy of a squared function by a constant times the integral of its squared gradient.
Compactly supported smooth function: An infinitely differentiable function that is zero outside a compact set.

Keywords : Logarithmic Sobolev inequality · Log-concave measures · Dimension-free bounds

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures

23/09/2026

To be verified

We prove that every centered log-concave probability measure with a Lebesgue density on ℝⁿ and linear subgaussian parameter a satisfies $Ent_{μ} (f^{2}) \leq C a^{2} \int ∣ D f ∣^{2} d μ$ for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

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