View source for The strong isotropic constant conjecture

'''The strong isotropic constant conjecture''' asks whether, in each dimension ''n'', simplices maximize the isotropic constant ''L''<sub>''K''</sub> = (det Σ<sub>''K''</sub> / |''K''|<sup>2</sup>)<sup>1/(2''n'')</sup> among convex bodies ''K'' in '''R'''<sup>''n''</sup>, where Σ<sub>''K''</sub> 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''<sub>''K''</sub>. Klartag proved a bound of order ''n''<sup>1/4</sup>; 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'''<sup>''n''</sup> satisfies ''L''<sub>''K''</sub> ≤ (''n''!)<sup>1/''n''</sup> / ((''n'' + 1)<sup>(''n''+1)/(2''n'')</sup> √(''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'''<sup>''m''</sup> 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''<sup>∘</sup>| ≥ (''n'' + 1)<sup>''n''+1</sup>/(''n''!)<sup>2</sup> 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''<sub>''m''</sub>√Δ(''f'') in Wasserstein distance ''W''<sub>2</sub> 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''<sub>''K''</sub> &lt; 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''<sub>''K''</sub>, as solved by the manuscript, noting that the source definition of ''L''<sub>''K''</sub> 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''<sup>2</sup>) holds with a slack of only about 7 × 10<sup>−6</sup> 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]]