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A sharp entropy bound and the simplex inequality for isotropic constants
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Abstract
We prove the strong isotropic constant conjecture: in each dimension, simplices are the unique maximizers of the isotropic constant among convex bodies. We also prove the sharp entropy bound $h(f)\ge m+\tfrac12\log\det\mathop{\mathrm{Cov}}\nolimits (f)$ for every log-concave probability density on ℝm, with equality precisely for invertible affine images of products of one-sided exponential laws.
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Follow-up works
- Haozhou Zhang, Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture (arXiv:2610.10979, v1 7 Oct 2026, v2 9 Oct 2026). Version 2 adds an appendix (Theorem A.1, "Makai from Strong Slicing") that derives Makai's reverse isodiametric conjecture, including its simplex equality case, from Theorem 1.1 of this paper; the author's note says they have "strong confidence in OpenAI's proof" but take no responsibility for that external result.
Possibly relevant literature (found by the agent, not checked)
- Shibing Chen, Yuanyuan Li, Dongmeng Xi, Zhe-Feng Xu, The Mahler Conjecture in Three Dimensions (arXiv:2605.09334, v1 10 May 2026). Proves the nonsymmetric Mahler inequality in dimension three with its equality case, which the paper obtains in every dimension as a corollary via Klartag (2018).
- Arnaud Marsiglietti, Victoria Kostina, A lower bound on the differential entropy of log-concave random vectors with applications (arXiv:1704.07766, 2017; DOI 10.3390/e20030185). Gives earlier non-sharp lower bounds on the differential entropy of log-concave random vectors in terms of moments, the quantity that Theorem 1.2 bounds sharply.
Digests and expositions
- Gradient Damage, Simplex Conjecture for Isotropic Constants: OpenAI Claims a Proof (video, 2026-10-16). This short animation illustrates the manuscript's simplex-maximizer claim by comparing isotropic constants of 22 three-dimensional solids, explicitly presenting an illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 101
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026,
author = {{OpenAI}},
title = {{A sharp entropy bound and the simplex inequality for isotropic constants}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026/isotropic-simplex.pdf}{OAI:A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
The scalar certificate checker uses only Python 3's standard library and exact rational arithmetic. From this paper's root directory, run:
python3 verification/verify_scalar.py
A successful run exits with status zero and ends with ALL CHECKS PASSED.
It checks the rational bounds used in the scalar estimates; the analytic
inputs are proved in the paper's appendix.
Family 101: The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
This is the only paper in this family.