Browse › Convex and metric geometry › 093 Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
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Abstract
We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies $\mathop{\mathrm{Ent}}\nolimits _\mu(f^2)\le Ca^2\int|Df|^2\,d\mu$ 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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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Pierre Bizeul, Boaz Klartag, Joseph Lehec, Presenting a proof of the Kannan-Lovász-Simonovits conjecture (arXiv:2610.05474, 4 October 2026). Claims a dimension-free Poincaré bound for all isotropic log-concave measures, the Poincaré-level analogue of the paper's log-Sobolev question, with Bizeul (who posed the conjecture this paper addresses) as a co-author; not in the paper's bibliography. Its acknowledgements state that most of its proofs were found by ChatGPT.
- Hong-Bin Chen, Sinho Chewi, Jonathan Niles-Weed, Dimension-free log-Sobolev inequalities for mixture distributions (arXiv:2102.11476, 2021). Proves dimension-free log-Sobolev inequalities for Gaussian convolutions of compactly supported measures, related to the paper's step of working with Gaussian convolutions $\mu_s=\mathrm{Law}(X+\sqrt s\,G)$ of compactly supported log-concave laws; not in the paper's bibliography.
Digests and expositions
- Latent Seminar, From subgaussian marginals to logarithmic Sobolev inequalities (6 October 2026). Explains the claimed inequality, its concentration consequences, and the prediction, tensor, and integrated-information stages of the argument, stating that it does not independently verify the complete proof. [family-level]
- Orply / Perplexity, Uniform Sub-Gaussian Projection Tails Yield Dimension-Free Log-Sobolev Bounds (9 October 2026). Summarizes the marginal hypothesis and the manuscript's contradiction argument through Gaussian prediction and mutual information integrated over observation scales. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 093
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 093: 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.
This is the only paper in this family.