Result 360, Differential geometry

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.

Lean formalization Proof

The bigger picture

Why it matters

Optimal transport asks how to move one distribution of mass into another at minimum cost. These manuscripts link a curvature condition to geometric convexity and continuity of the resulting transport on curved spaces.

What changes?

The first manuscript reports that weak Ma-Trudinger-Wang curvature, a condition on squared-distance transport geometry, forces every tangent injectivity domain to be convex. Such a domain consists of initial velocities describing shortest paths from a point before they reach its cut locus; convexity means straight segments between these velocities stay inside. The claim covers every smooth, connected, compact manifold without boundary of dimension at least two. It allows conjugate cut points and requires no density assumptions.

What does that help mathematicians do?

For squared-distance transport on any fixed manifold satisfying these assumptions, the second manuscript reports a common Hölder estimate for optimal maps and their inverses across all measurable probability densities with fixed positive lower and upper bounds. The maps have representatives that are continuous bijections with continuous inverses. Hölder control bounds changes in output by a power of changes in input. Thus researchers obtain uniform continuity control from geometry and density bounds alone, without assuming smooth densities.

Are there practical applications?

The immediate value is foundational: the convexity claim resolves Villani's conjecture in the stated setting, while the transport claim identifies conditions preventing discontinuities even for rough mass distributions. This strengthens the mathematical basis for studying squared-distance transport on curved spaces. The supplied abstracts do not establish a computational method or practical performance gains.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Global Support and Convex Injectivity Domains under Weak MTW

September 25, 2026 24 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Uniform Bi-Holder Transport from Weak MTW

September 25, 2026 41 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/360.md.

Weak MTW curvature gives convexity and regular optimal transport

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

On a compact connected smooth Riemannian manifold of dimension at least two, the formalized result proves that weak MTW curvature implies convexity of every open tangent injectivity domain. Prior convexity or nonfocality is not assumed. Additional formalized results give global support at ordinary subgradients, compact convex lifted gap sections with the stated diameter bound, and scale and local-injectivity estimates for finite target families. The latter finite-family result does not cover the paper's arbitrary-potential local-injectivity assertion.

The formalized result gives uniform bi-Hölder optimal transport for squared-distance cost on each fixed compact connected smooth Riemannian manifold of dimension at least two satisfying weak MTW. For probability densities between fixed positive lower and finite upper bounds, one exponent and constant work for every pair of densities. The optimal map is a homeomorphism, is unique almost everywhere, and both it and its inverse satisfy the global Hölder bound. The constants are uniform over the density class on the fixed manifold, not over varying metrics.

Comparator links

Result Comparator statement
Convex injectivity domains under weak MTW WeakMTWGlobalSupport.lean
Uniform bi-Hölder optimal transport BiholderTransport.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.