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Weak MTW curvature gives convexity and regular optimal transport

Proves weak MTW curvature makes every injectivity domain convex, as Villani conjectured, and transport maps Hölder.

2 papers September 25, 2026 Explained by AI

Optimal transport asks for the cheapest way to move one spread of mass to another, with cost equal to half the squared distance. On a curved space, a curvature condition called weak MTW controls this cost where it is smooth. The first paper proves that this condition makes every injectivity domain convex, as Villani conjectured. The second proves that the best transport map and its inverse are Hölder continuous, with one constant for a whole class of densities. Lean, a computer proof checker, has formal statements of both main results.

First step
2005
Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang, a curvature condition for optimal transport regularity
Significance
Major
It settles Villani's conjecture without nonfocality and gives class-uniform transport regularity, both with Lean statements.
Trust
Formal proof listed
Field
Differential geometry
2 papers

The explanation

Take a flat square torus of side 1, like a video-game screen where leaving one edge brings you back on the opposite edge. Stand at a point x and walk straight with velocity v. If v = (0.4, 0.3), you travel length 0.5 and no shorter path exists. If v = (0.6, 0), you end 0.4 away on the other side, so your straight path was not the shortest. The velocities whose straight paths stay shortest a little beyond time one form an open set called the injectivity domain. On this torus it is the open square where both coordinates lie strictly between −1/2 and 1/2. That square is convex: the midpoint of (0.45, 0.1) and (0.1, 0.45) is (0.275, 0.275), still inside.

Optimal transport moves one spread of mass to another at least total cost. Here the cost of moving from x to y is half the squared distance. The Ma–Trudinger–Wang condition, MTW for short, is a curvature condition on this cost. The weak version only asks for a sign in directions perpendicular to each other. Villani proposed that weak MTW forces every injectivity domain to be convex. Figalli, Gallouët and Rifford proved this for nonfocal manifolds, where shortest paths stop being shortest before nearby paths refocus. The obstacle is that the cost stops being smooth along some proposed paths. Weak MTW controls only directions perpendicular to the path.

The first paper proves the conjecture on every compact connected manifold of dimension at least two without boundary. Points where nearby paths refocus are allowed. It also shows that every ordinary subgradient of a cost potential is a shortest velocity with a global supporting mountain. No assumption on densities is used.

The second paper uses this to control the transport map. Fix densities with upper bound Λ and positive lower bound λ. The best map T and its inverse then obey d(T(x), T(x′)) ≤ C d(x, x′)^α. This is Hölder continuity. For example, with α = 1/2, points 0.01 apart go to points at most 0.1 C apart. The constants α and C work for the whole class of densities on a fixed manifold. The exponent is found by contradiction, so no explicit value is given. With an earlier result of Figalli, Rifford and Villani, this gives an equivalence between weak MTW and continuity of transport.

How sure is this? Lean, a computer proof checker, has formal statements of both main results. So the papers prove them, as far as the Lean statements match the papers. One local-injectivity statement is covered only for finite target families. No specialist review is published.

What it doesn't show. This does not give an explicit Hölder exponent, or constants that stay uniform when the metric changes. The Lean scope also omits the paper's arbitrary-potential local-injectivity assertion.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major, with Lean statements of both main results. The Hölder exponent is not explicit, and the constants depend on the fixed manifold. No specialist review is published.

Words used

Optimal transport
Moving one spread of mass to another at the least total cost.
Injectivity domain
The set of starting velocities whose straight paths stay shortest a little beyond time one.
Weak MTW condition
A sign condition on a fourth derivative of the squared-distance cost, asked only in perpendicular directions.
Convex
Containing the whole straight segment between any two of its points.
Nonfocal
A manifold where shortest paths stop being shortest strictly before nearby paths refocus.
Hölder continuous
Moving close points to close points, with distance controlled by a power of the original distance.
Subgradient
A slope that supports a function from below at a point, even where it has a corner.

The original papers (2)

Lean notes

All quoted lines (12)
  1. Its geodesics remain minimizing beyond time one.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  2. The weak Ma–Trudinger–Wang condition controls the cost Hessian only in directions perpendicular to a velocity segment.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  3. Villani proposed that weak MTW should force convexity of every tangent injectivity domain.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  4. Figalli, Gallou\"et and Rifford proved the implication for nonfocal manifolds, where minimizing geodesics reach cut strictly before their first conjugate point.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  5. The cost may cease to be smooth along a proposed velocity segment, and weak MTW controls only directions perpendicular to that segment.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  6. 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.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  7. … removes this restriction: conjugate cut endpoints are allowed in every dimension n≥2n\geq2.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  8. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain.
    Global Support and Convex Injectivity Domains under Weak MTW Back to text
  9. The almost-everywhere unique optimal map from ρ0\vol\rho_0\vol to ρ1\vol\rho_1\vol for the cost cc has a homeomorphic representative T~:M→M\widetilde T:M\to M, and d(T~(x),T~(x′))≤Cd(x,x′)α,d(T~−1(y),T~−1(y′))≤Cd(y,y′)αd(\widetilde T(x),\widetilde T(x'))\leq C d(x,x')^\alpha, \qquad d(\widetilde T^{-1}(y),\widetilde T^{-1}(y')) \leq C d(y,y')^\alpha for all x,x′,y,y′∈Mx,x',y,y'\in M.
    Uniform Bi-Holder Transport from Weak MTW Back to text
  10. The constants are common to the full density class on the fixed manifold.
    Uniform Bi-Holder Transport from Weak MTW Back to text
  11. We obtain a positive exponent by contradiction; no explicit exponent or uniformity over varying metrics is asserted.
    Uniform Bi-Holder Transport from Weak MTW Back to text
  12. Combined with their necessity result, it yields the weak-MTW–transport-continuity equivalence, with the forward direction strengthened to the class-uniform estimates above.
    Uniform Bi-Holder Transport from Weak MTW Back to text

Other sources

  1. openai/math, Lean scope note for family 360 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.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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