Delta-convex structure of the singular set of distance functions (Q6587587)

The setting of this paper is that of a complete Finsler manifold. The authors study the singular set of the distance function to an arbitrary closed subset (that is, the set of points where this function is not differentiable). They show that such a singular set is a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In the case of dimension two, they show that the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. The results are new even in the case where the ambient manifold is an Euclidean space, and they generalize previous results on this question. Furthermore, the authors show that these results are optimal.