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An approximation lemma about the cut locus, with applications in optimal transport theory. (English) Zbl 1172.53022
Summary: A path in a Riemannian manifold can be approximated by a path meeting only finitely many times the cut locus of a given point. The proof of this property uses recent works of Itoh-Tanaka and Li-Nirenberg about the differential structure of the cut locus. We present applications in the regularity theory of optimal transport.

MSC:
53C20 Global Riemannian geometry, including pinching
35B65 Smoothness and regularity of solutions to PDEs
49N99 Miscellaneous topics in calculus of variations and optimal control
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