Zero Probability of the Cut Locus of a Fr\'echet Mean on a Riemannian Manifold

Abstract

We show that the cut locus of a Fr\'echet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fr\'echet functions and we discuss extensions of the statement to Fr\'echet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces.