Proved
A logarithmic bound (Klartag 2023) ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Bound · version 1
A logarithmic bound
Statement
There exists C > 0 such that for every n ≥ 2, every isotropic log-concave probability measure μ on ℝⁿ has C_P(μ) ≤ C log n.
Assumptions
- Isotropic log-concave probability measures; C_P is the Poincaré constant.
- Reported in the 2026 paper’s account of earlier work.
Scope
A proved dimension-dependent bound; it does not give one constant for every dimension.
Evidence
Source reports
Connections
In this problem · publication v5
Kannan–Lovász–Simonovits conjecture →
Related result · Proved
An earlier proved bound whose growth in n the conjecture would remove.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Presenting a proof of the Kannan–Lovász–Simonovits conjecture →
- §1, paragraph on stochastic localization and (8); references [12], [22], [23], [28] → · Original passage ↗
Recorded wording
For isotropic log-concave measures, Lee and Vempala obtained C_P(μ) ≤ C√n (Ann. Math. 2024); Chen established a sub-polynomial bound (GAFA 2021); Klartag and Lehec proved a poly-logarithmic bound (GAFA 2022); and Klartag proved C_P(μ) ≤ C log n (2023).
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 · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
A logarithmic bound (Klartag 2023) ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →
Record details
- Claim ID
- claim-kls-klartag-log
- Statement version
- 1
- SHA-256
- 69d4996cf73f059fac07fe9f29dd11b8090ae4a16b2c11a1035524f637e50e75