OP-00736 Solution claimed

Villani’s convex-injectivity-domain conjecture

Let $(M,g)$ be a smooth compact connected Riemannian manifold without boundary, with geodesic distance $d$ and exponential map $\exp$. For $x \in M$ let $$I_x = \bigl\{v \in T_xM : d(x, \exp_x(av)) = a|v|_g \text{ for some } a > 1\bigr\}$$ be the tangent injectivity domain at $x$, and for $v \in I_x$ and $\xi, \eta \in T_xM$ let $$S_{x,v}(\xi,\eta) = -\frac32 \left.\partial_s^2 \partial_t^2 \frac{d^2\bigl(\exp_x(t\xi), \exp_x(v + s\eta)\bigr)}{2}\right|_{s=t=0}$$ be the Ma–Trudinger–Wang tensor. The weak Ma–Trudinger–Wang condition is $S_{x,v}(\xi,\eta) \ge 0$ for all $x \in M$, $v \in I_x$ and $\xi, \eta \in T_xM$ with $\langle \xi, \eta \rangle_g = 0$. Since $S_{x,0}$ evaluated on orthonormal $\xi, \eta$ is the sectional curvature of their plane, the condition forces nonnegative sectional curvature (Loeper). Loeper and Villani proved that every $I_x$ is uniformly convex on nonfocal manifolds satisfying a strict form of the condition, and Figalli, Gallouët and Rifford proved that every $I_x$ is convex on nonfocal manifolds satisfying the weak condition.

Problem. Assume the weak Ma–Trudinger–Wang condition. Must $I_x$ be convex for every $x \in M$? No hypothesis excluding focal points on the cut locus is imposed.

Source: Villani (2011); stated as Villani's conjecture by Figalli, Gallouët and Rifford (2015), after their Theorem 1.7.

Imported from github.com/MColbrook/AIM.

Classification

Primary: DG — Differential Geometry
Secondary: AP, OC
Keywords: AIM Open Applied Problems, injectivity domains, Ma–Trudinger–Wang condition, optimal transport

References

@article{villani2011,
  author  = {Villani, C{\'e}dric},
  title   = {Regularity of optimal transport and cut locus: from nonsmooth analysis to geometry to smooth analysis},
  journal = {Discrete Contin. Dyn. Syst.},
  volume  = {30},
  number  = {2},
  year    = {2011},
  pages   = {559--571},
  doi     = {10.3934/dcds.2011.30.559},
}
@article{figalli2015,
  author  = {Figalli, Alessio and Gallou{\"e}t, Thomas O. and Rifford, Ludovic},
  title   = {On the convexity of injectivity domains on nonfocal manifolds},
  journal = {SIAM J. Math. Anal.},
  volume  = {47},
  number  = {2},
  year    = {2015},
  pages   = {969--1000},
  doi     = {10.1137/140961821},
}
@article{loeper2009,
  author  = {Loeper, Gr{\'e}goire},
  title   = {On the regularity of solutions of optimal transportation problems},
  journal = {Acta Math.},
  volume  = {202},
  number  = {2},
  year    = {2009},
  pages   = {241--283},
  doi     = {10.1007/s11511-009-0037-8},
}
@article{loeper2010,
  author  = {Loeper, Gr{\'e}goire and Villani, C{\'e}dric},
  title   = {Regularity of optimal transport in curved geometry: the nonfocal case},
  journal = {Duke Math. J.},
  volume  = {151},
  number  = {3},
  year    = {2010},
  pages   = {431--485},
  doi     = {10.1215/00127094-2010-003},
}
@misc{aim-villani-mtw-convex-injectivity-domains,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Villani’s convex-injectivity-domain conjecture}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/394-villani-mtw-convex-injectivity-domains.md},
  note         = {Entry 394 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Global Support and Convex Injectivity Domains under Weak MTW v1 GitHub ↗

OpenAI

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 …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-16	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00736, "Villani’s convex-injectivity-domain conjecture", clopen.solutions.