Aftermath bot: New page from public sources: the simplex inequality for isotropic constants and four public follow-ups

2026-10-23T11:03:36Z

New page from public sources: the simplex inequality for isotropic constants and four public follow-ups

New page

'''The strong isotropic constant conjecture''' asks whether, in each dimension ''n'', simplices maximize the isotropic constant ''L''''K'' = (det Σ''K'' / |''K''|2)1/(2''n'') among convex bodies ''K'' in '''R'''''n'', where Σ''K'' is the covariance matrix of the uniform distribution on ''K''. It is the sharp form of Bourgain's slicing problem. A manuscript dated 5 October 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claims the conjecture in every dimension, together with a sharp entropy lower bound for log-concave densities. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.

== Background ==
* As recounted in the introduction of the manuscript:
** The slicing problem, arising in Bourgain's work, asks for a dimension-free upper bound on ''L''''K''. Klartag proved a bound of order ''n''1/4; Eldan's stochastic localization led through subpolynomial and polylogarithmic estimates to Klartag and Lehec's dimension-independent bound using Guan's covariance estimate. A universal bound does not identify the extremal value or an extremizing body.
** In the plane, Campi, Colesanti and Gronchi proved the simplex bound and Saroglou established uniqueness of the maximizer. In higher dimensions, Rademacher proved that a simplicial polytope maximizing the isotropic constant must be a simplex, and Kipp obtained restrictions on local maximizers in terms of decomposability.
** Bobkov and Madiman related entropy bounds for log-concave measures to the slicing problem. Fradelizi and Marín Sola formulated the sharp entropy statement below and proved its equivalence, across dimensions, with the simplex bound; Melbourne, Nayar and Roberto proved it in dimension one.

== The original results ==
* '''Claim (the simplex inequality).''' Theorem ''intro:simplex'' of ''A sharp entropy bound and the simplex inequality for isotropic constants'' states that for every ''n'' ≥ 1 every convex body ''K'' in '''R'''''n'' satisfies ''L''''K'' ≤ (''n''!)1/''n'' / ((''n'' + 1)(''n''+1)/(2''n'') √(''n'' + 2)), with equality if and only if ''K'' is a simplex.
* '''Claim (the sharp entropy bound).''' Theorem ''intro:entropy'' states that every log-concave probability density ''f'' on '''R'''''m'' satisfies ''h''(''f'') ≥ ''m'' + ½ log det Cov(''f''), with equality exactly for invertible affine images of products of independent one-sided exponential laws.
* '''Consequence, as stated by the manuscript.''' Through Klartag's implication from the sharp simplex bound, the theorem gives the nonsymmetric Mahler inequality |''K''| |''K''∘| ≥ (''n'' + 1)''n''+1/(''n''!)2 in every dimension.
* ''Method, as described:'' the proof transports the standard Gaussian measure to a regularized density by Brenier's map, parametrizes its Hessian through the scalar transport from a Gaussian to a mean-one exponential variable, normalizes the resulting matrix field to have Gaussian mean zero, and closes an energy estimate that keeps a term involving pairs of distinct eigenvalues. One-variable bounds are certified in an appendix with an exact-rational verifier.
* '''Sources.''' At commit <code>1581d4c</code>: [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026 A sharp entropy bound and the simplex inequality for isotropic constants].
* '''Formalization.''' The repository's Lean documentation has no scope note for this family at that commit.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 23 October 2026)
|-
| 2026-10-07 || mxym, ''Quantitative compatibility of nonlinear matrix Jacobians'', with two companion manuscripts on orthogonal tensor rigidity ([https://github.com/mxym/math/tree/c86f853d067e2105c508d6c25e2739593c166199/notes/global-orthogonal-tensor-rigidity mxym/math, notes/global-orthogonal-tensor-rigidity]) || Uses (conditional stability) || Assuming the transport remainder stated in the manuscript's smooth entropy bound, as an explicitly imported hypothesis, the note derives square-root stability: for smooth strongly log-concave densities with small entropy deficit Δ(''f'') = ''h''(''f'') − ''m'' − ½ log det Cov(''f''), and for all full-dimensional log-concave densities when the manuscript's approximation lemma is used as well, the whitened law of ''f'' lies within ''C''''m''√Δ(''f'') in Wasserstein distance ''W''2 of a rotated product of centered mean-one exponential laws. An unconditional one-dimensional truncated-exponential family shows that no power larger than ½ is possible. The note attributes its exact compatibility argument to the manuscript's rigidity lemma. || Repository note prepared with AI assistance and self-reviewed; it states that it does not independently certify the imported remainder. A Lean file checks seven algebraic lemmas, not the analytic results.
|-
| 2026-10-08 || [https://github.com/atimics/math2/blob/2c0f00528a8bb375d02bd48a513062254a53c285/docs/COROLLARIES.md atimics/math2], entry GE-4 of <code>docs/COROLLARIES.md</code> (GitHub, commit <code>2c0f005</code>) || Uses (written corollary) || From the simplex inequality, by elementary bounds on ''n''!, the simplex value is below 1/''e'' in every dimension and tends to 1/''e''; hence ''L''''K'' < 1/''e'' for every convex body in every dimension, and 1/''e'' is the best dimension-free constant. || Catalogue of conditional corollaries in a public repository; AI-assisted.
|-
| 2026-10-13 || M. Colbrook and contributors, [https://github.com/MColbrook/AIM AIM — Open Applied Problems] (GitHub), [https://github.com/MColbrook/AIM/blob/8729eab98bafe409b66c5e39253bf56569a62327/problems/543-simplex-maximal-isotropic-constant.md problem 543] || Catalogue || The collection records its problem 543, the sharp simplex bound for ''L''''K'', as solved by the manuscript, noting that the source definition of ''L''''K'' matches, that no symmetry or polytope restriction appears, and that the uniqueness of the equality case goes beyond the target. || The status review is described as "an AI statement-level review of a published manuscript, not a fresh independent audit of its complete proof or a Lean kernel replay". The collection is not affiliated with the American Institute of Mathematics.
|-
| 2026-10-16 || J. Jardine, Lattice24, referee review [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_117.json rank_117] of the manuscript in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] || Review (AI-assisted audit) || Reports no gap. It records one minor reservation: the appendix's tail estimates for the one-variable bounds were not audited line by line; instead the reviewer rechecked the bounds on a grid in high precision and ran the authors' exact-rational verifier, noting that the bound ''p''(''a'') ≥ 1/(7 + ''a''2) holds with a slack of only about 7 × 10−6 near its worst point. || Part of an audit of the collection's manuscripts that the audit describes as complete.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

The strong isotropic constant conjecture - Revision history

Aftermath bot: New page from public sources: the simplex inequality for isotropic constants and four public follow-ups