Villani's conjecture on convexity of injectivity domains under the MTW condition
Let be a Riemannian manifold, let be its Riemannian distance, and let the squared-distance cost be . The cost satisfies the MTW condition if the MTW inequality holds for all orthogonal vector-covector pairs as in the definition above.
Villani's conjecture. If the squared-distance cost satisfies the MTW condition, then all injectivity domains of are convex.
This conjecture concerns the geometric consequences of the MTW condition in optimal transport. The surrounding discussion presents it as a prominent open question in the regularity theory; no resolution is given here.
References
Primary source
Gabriel Khan and Jun Zhang, “When Optimal Transport Meets Information Geometry”, arXiv:2206.14791 (2022).
Progress summary
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Solutions 1
RemarkAI-assistedClaimed by OpenAI. Claims convexity of tangent injectivity domains under weak Ma–Trudinger–Wang curvature on smooth connected compact boundaryless Riemannian manifolds of dimension at least two, allowing conjugate cut points and requiring no density hypotheses.See full solution
Claimed by OpenAI. Claims convexity of tangent injectivity domains under weak Ma–Trudinger–Wang curvature on smooth connected compact boundaryless Riemannian manifolds of dimension at least two, allowing conjugate cut points and requiring no density hypotheses.
Scope relative to this problem: The source uses smooth connected compact boundaryless Riemannian manifolds of dimension at least two and the weak MTW condition for squared distance. It claims convexity of each tangent injectivity domain even with conjugate cut points. It does not extend this manuscript to arbitrary noncompact or nonsmooth spaces mentioned by a broader reading of the target.
GitHub repository: https://github.com/openai/math
- OpenAI-360-01-Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW.pdfOpen