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Uniform Bi-Holder Transport from Weak MTW
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Abstract
We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.
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Notes
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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.
Formalization
- OpenAI math release, Weak MTW curvature gives convexity and regular optimal transport (Lean scope notes) (lean/docs/360.md, 2026). Describes the release's Lean formalization of uniform bi-Hölder optimal transport for squared-distance cost on a fixed weak-MTW manifold, with constants uniform over the density class but not over varying metrics. [family-level]
Related prior literature mentioned by others
- Alessio Figalli, Thomas O. Gallouët and Ludovic Rifford, On the convexity of injectivity domains on nonfocal manifolds (SIAM Journal on Mathematical Analysis 47, 969–1000, 2015). Identified as prior work in AIProofWatch's discussion of the convexity manuscript; its Theorem 1.7 establishes the weak-MTW convexity implication under nonfocality, the hypothesis the family's geometric input claims to remove. [family-level]
Possibly relevant literature (found by the agent, not checked)
- N. Guillen and J. Kitagawa, The local geometry of maps with c-convex potentials (arXiv:1212.4865, 2012). Proves local Hölder-type regularity of optimal maps between densities bounded above and below under a three-derivative condition equivalent to weak MTW for $C^4$ costs, bearing on the introduction's discussion of density-controlled local Hölder estimates; not cited in the paper's bibliography.
Digests and expositions
- Latent Seminar, Weak MTW curvature: convex injectivity domains and regular optimal transport (expository article, 2026-10-07). Explains the geometric continuation argument and the companion's section-to-Hölder reduction, spectral estimates and determinant argument, stating that it does not independently verify the complete proofs. [family-level]
Elsewhere
- Emergent Mind: family 360
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026,
author = {{OpenAI}},
title = {{Uniform Bi-H{\"o}lder Transport from Weak MTW}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026/paper.pdf}{OAI:Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/360.md. The Lean statements have not been checked by people against the paper's claims.
Family 360: Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.
Other papers in this family:
- Global Support and Convex Injectivity Domains under Weak MTW · September 25, 2026