A uniform Poincaré bound for isotropic log-concave measures
Statement
There exists C > 0 such that for every n ≥ 1 and every isotropic log-concave probability measure μ on Rn, Varμ(f)≤C∫∣∇f∣2dμ for every locally Lipschitz real f for which the variance and gradient energy are finite.
Assumptions
Isotropic: mean zero and covariance I.
Log-concave density and positive finite dimension; the same C works for all dimensions.
Scope
One shared mathematical assertion with separately attributed October proof proposals. Neither paper receives an independent-review label from this curation.
Explanation
Proof outlines · 2
Song second-revision account
A proposed dimension-free argument
The second revision proposes a universal bound, strengthening the first revision’s iterated-logarithm dependence.
This is a proposed proof under assessment; the earlier quantitative assertion remains a separate historical Claim.
Proposed:Uniform bound proposed by Song and Zhang in revision two ↗ · source dated Oct 4, 2026 · observed Oct 13, 2026. The later manuscript proposes a constant independent of dimension; independent mathematical assessment is pending.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
A separately authored proof account addresses the same uniform Poincaré assertion using a criterion from Song’s work.
The dependency on Song’s criterion means this account is not independent verification of that criterion.
Proposed:Uniform bound proposed by Bizeul, Klartag and Lehec ↗ · source dated Oct 4, 2026 · observed Oct 13, 2026. Theorem 1.1 asserts a universal Poincaré bound. No formal execution or independent mathematical assessment is recorded here.
Independent mathematical assessment of this argument remains separate from the statement’s attributed status.
The second revision proposes a uniform O(1) bound, strengthening the first revision’s slowly dimension-dependent estimate.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Zhao Song · Xinzhi Zhang · Preprint · Publication date Oct 4, 2026
Revision & attribution
Source revision: arXiv:2610.01447v2. Observed Oct 13, 2026.
Song and Zhang credit GPT-6 Astra, GPT-5.6 Sol and Claude Fable 5/5.1 for extensive proof exploration and checking, with authors directing priorities and retaining responsibility.
Proposes a dimension-free Poincaré bound for isotropic log-concave measures through cumulant estimates, localization and suspension.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Pierre Bizeul · Boaz Klartag · Joseph Lehec · Preprint · Publication date Oct 4, 2026
Revision & attribution
Source revision: 2026-10-04. Observed Oct 13, 2026.
The authors credit ChatGPT with many proof ideas and arguments, and credit the suspension idea and exposition to the human authors. Their approach uses a criterion of Song and Zhang; these are distinct papers with a mathematical dependency, not wholly independent proofs.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.