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

What OpenAI claims

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.

Paper

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

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove that every centered log-concave probability measure with a Lebesgue density on ℝn and linear subgaussian parameter a satisfies Ent _μ(f²) ≤ Ca²∫|Df|² dμ for compactly supported smooth f, with one universal constant C. This resolves positively the dimension-free logarithmic Sobolev conjecture for subgaussian log-concave measures.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.