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Global Support and Convex Injectivity Domains under Weak MTW
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Abstract
We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani's conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.
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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 convexity of every open tangent injectivity domain under weak MTW, plus global support at ordinary subgradients and convex lifted gap sections; it states that the formalized local-injectivity estimate covers only finite target families, not the paper's arbitrary-potential assertion. [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)
- G. Loeper and N. S. Trudinger, Weak formulation of the MTW condition and convexity properties of potentials (arXiv:2007.02665, 2020). Gives a geometric proof that locally c-convex potentials are globally c-convex for $C^2$ costs, which may be relevant to the global supporting inequality for ordinary subgradients in Theorem 1.2(i).
- A. Figalli, L. Rifford and C. Villani, Nearly round spheres look convex (Amer. J. Math. 134(1), 109–139, 2012, doi:10.1353/ajm.2012.0000). Earlier work on convexity of injectivity domains for metrics close to the round sphere, a known class of examples relevant to Theorem 1.1 that the paper's introduction does not discuss.
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
- VibeMathed: Problem record
- Emergent Mind: family 360
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026,
author = {{OpenAI}},
title = {{Global Support and Convex Injectivity Domains under Weak MTW}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf}{OAI:Global-Support-and-Convex-Injectivity-Domains-under-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:
- Uniform Bi-Holder Transport from Weak MTW · September 25, 2026