Methods and Applications of Analysis

Volume 15 (2008)

Number 2

An Approximation Lemma about the Cut Locus, with Applications in Optimal Transport Theory

Pages: 149 – 154



Alessio Figalli

Cedric Villani


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.


Cut locus; optimal transport; co-area formula

2010 Mathematics Subject Classification

35B65, 49Q20, 53C20

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