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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.

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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