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Sharp One-Third Stability of Brenier Maps
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Abstract
For the uniform probability measure ρ on a compact convex body in ℝd, d ≥ 2, quadratic optimal transport maps are one-third Hölder continuous in $L^2(\rho)$ with respect to the target's 2-Wasserstein distance. The constant is uniform over all targets supported in a fixed compact set, and the exponent is optimal. A three-atom family on a fixed cube disproves Letrouit's conjectured uniform square-root estimate.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- F.-U. Caja-Lopez, M. G. Delgadino, J. Kitagawa, Stability of optimal transport maps and second variation of the 2-Monge–Kantorovich distance (arXiv:2605.24232v2, revised 19 Oct 2026). This paper is in the manuscript's bibliography as v1; v2 adds a citation of "the manuscript released by OpenAI" as showing that their bound (1.3) holds with sharp exponent α=1/3 for a uniform source on a convex body.
- Songbo Wang, Xiaozhen Wang, Quantitative stability of optimal transport via regularization (arXiv:2610.09640, 7 Oct 2026). Proves W₂-Hölder map stability for L^p source densities via regularized Brenier potentials and cites "the OpenAI preprint" for the sharp W₂^{1/3} estimate for uniform sources on convex bodies; the authors also disclose a dialogue with OpenAI Codex in an AI-use section.
- Bang-Xian Han, Zhuo-Nan Zhu, Stability of optimal transport on sub-Riemannian manifolds (arXiv:2610.09688, 7 Oct 2026). Proves a W₁^{1/4} map estimate on Riemannian and fat sub-Riemannian manifolds, describes the OpenAI preprint as a concurrent work whose example "disproves the square-root conjecture", and notes that its own W₁ estimate does not give a W₂^{1/3} bound in its settings.
- Anonymous, Exact sensitivity certificates for planar transport maps: coefficient uncertainty, a convergent box refinement and a sharp three-atom calibration (Evidence Press, version 0.2.0-candidate, 2026-10-07; DOI 10.5281/zenodo.23205731). This unrefereed work extends OpenAI's three-atom construction with exact planar sensitivity formulas and coefficient-box certificates, explicitly stating that it neither uses nor validates the general one-third theorem. [AI-generated per source]
Formalization
- OpenAI, Sharp one-third stability of Brenier maps — Lean scope note (lean/docs/374.md, OpenAI math release, 2026-10-06). States that the Lean development covers both the uniform W₂^{1/3} bound for targets in a fixed compact set and the three-atom sharpness on a cube, with comparator statement
lean/ComparatorChallenges/Brenier.lean. [AI-generated per source] - Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit at commit fd4aeeb2, 2026-10-10). The audit reports that the Brenier challenge passed Lean and nanoda checking, that all twelve displayed definition bodies matched, and that its transfer check to the displayed statement passed.
Related prior literature mentioned by others
- Cyril Letrouit, Unstable optimal transport maps (arXiv:2510.13265, 15 Oct 2025; published in C. R. Math. 364 (2026)). Han–Zhu (arXiv:2610.09688) identify Conjecture 3 of this note as the square-root conjecture the manuscript disproves; the manuscript cites the Comptes Rendus version.
- Mohit Bansil and Jun Kitagawa, Quantitative stability in the geometry of semi-discrete optimal transport (arXiv:2002.02022, 2020-02-05; revised 2021-02-10). The Evidence Press follow-up identifies this work on Laguerre-cell stability under changing masses at fixed sites as related background for its extension of the paper's three-atom construction.
Possibly relevant literature (found by the agent, not checked)
- Jun Kitagawa, Cyril Letrouit, Quentin Mérigot, Stability of optimal transport maps on Riemannian manifolds (arXiv:2504.05412, 2025). Proves target-uniform quantitative stability for maps and Kantorovich potentials on Riemannian manifolds, with a sharpness-of-exponent example in v2; a comparison point for the manuscript's potential-then-gradient route to the 1/3 exponent and its sharpness claim.
- Yuanlong Ruan, Sharp conditioning and quantitative stability of optimal transport (arXiv:2610.07577, 2026-10-06). Section 12 derives common-source Brenier-map stability from transport-cost error bounds, including a compact-support W₁ quarter-power estimate for bounded source densities, for comparison with the manuscript's W₂ one-third estimate.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 374
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026,
author = {{OpenAI}},
title = {{Sharp One-Third Stability of Brenier Maps}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026/article.pdf}{OAI:Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/374.md. The Lean statements have not been checked by people against the paper's claims.
Family 374: Sharp one-third stability of Brenier maps
For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.
This is the only paper in this family.