TITLE: Applications of cut locus structure theorems
QUESTION [14 upvotes]: I am preparing a presentation discussing the structure of the cut locus on Riemannian and Finsler manifolds. Since my audience will consist mostly of applied mathematicians, I was hoping to include some nice applications to other (non-geometric) areas of mathematics which might appeal to the majority of the audience. While I'm vaguely aware that there are some applications to regularity theory, I do not understand these results. I am therefore looking for a nice introduction to this topic in addition to any other applications which might appeal to a group of applied mathematicians.

REPLY [3 votes]: Here is a possibility:

Villani, Cédric. "Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis." Discrete Contin. Dyn. Sys. 30.2 (2011): 559-571.

Abstract. In this survey paper I describe the convoluted links between the regularity theory of optimal transport and the geometry of cut locus.

Theorem 4.1 (convex Earth). [...]
  Informally speaking: assume that the Earth is very smooth, but not
  exactly round (there are hills etc.) The theorem says that if you draw a map of the Earth from one given point, it may not look round, but at least it will look convex"