Result 093, Convex and metric geometry

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

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.

Proof

The bigger picture

Why it matters

Can mild fluctuations in every direction control a distribution's more complicated behavior? This manuscript claims they can for a broad class of convex-shaped probability distributions, with a bound that does not worsen as the number of coordinates grows.

What changes?

The unreviewed manuscript reports a bound for every mean-zero probability measure with a Lebesgue density in any finite dimension whose log-density is concave. Assume every unit-direction projection is subgaussian, meaning its tails obey Gaussian-type bounds with a common scale a. For every smooth, compactly supported function, the entropy of its square is at most C times a squared times its mean squared gradient. The constant C is universal, independent of dimension and the particular measure. This is the claimed logarithmic Sobolev inequality.

What does that help mathematicians do?

Entropy here measures how unevenly a nonnegative quantity is distributed relative to its mean; the gradient measures how rapidly the test function changes. The result would upgrade control of linear projections to control of nonlinear test functions. In particular, it rules out families satisfying these assumptions with a fixed subgaussian scale but logarithmic Sobolev constants growing without bound as dimension increases.

Are there practical applications?

Its immediate value is foundational for high-dimensional probability and convex geometry. Researchers could use this dimension-independent entropy bound when studying fluctuations under the specified distributions, without paying an extra factor for the number of coordinates. Such inequalities connect to concentration estimates, but the abstract does not demonstrate a practical sampling method or computational speedup.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 23, 2026 65 pages

We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies Entμ(f2)≤Ca2∫∣Df∣2 dμ\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu 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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Cite (BibTeX) 
@misc{OAI:A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026,
  author = {{OpenAI}},
  title = {{A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf}{OAI:A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.