Delta-convex structure of the singular set of distance functions

TL;DR

This paper investigates the structure of the singular set of distance functions in Finsler manifolds, revealing it as a union of delta-convex hypersurfaces with optimal regularity properties, even in Euclidean spaces.

Contribution

It establishes the delta-convex structure of the singular set of distance functions in general Finsler manifolds, including Euclidean space, with optimal regularity results.

Findings

Singular set equals a countable union of delta-convex hypersurfaces with codimension two exceptions.

In dimension two, the singular set is a union of delta-convex Jordan arcs with isolated points.

Results are new and optimal in terms of regularity, even in standard Euclidean space.

Abstract

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.