The cut locus and distance function from a closed subset of a Finsler manifold

TL;DR

This paper characterizes the differentiability of the distance function from a closed subset in Finsler manifolds and describes the structure of the cut locus, revealing new insights even in Riemannian cases.

Contribution

It provides a new characterization of differentiable points of the distance function and establishes the structure of the cut locus in 2D Finsler manifolds, including a local tree structure.

Findings

Differentiability points characterized by the number of N-segments.

Cut locus in 2D Finsler manifolds is a local tree.

Intrinsic metric on the cut locus matches the induced topology.

Abstract

We characterize the differentiable points of the distance function from a closed subset $N$ of an arbitrary dimensional Finsler manifold in terms of the number of $N$-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset $N$, namely that it is a local tree, it is made of countably many rectifiable Jordan arcs except for the endpoints of the cut locus and that an intrinsic metric can be introduced in the cut locus and its intrinsic and induced topologies coincide. We should point out that these are new results even for Riemannian manifolds.