Delta-convex structure of the singular set of distance functions

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.

35 pages, 3 figures, final verson

References in corpus (3)

• The cut locus and distance function from a closed subset of a Finsler manifold
• Distance functions with dense singular sets
• The tangent cone, the dimension and the frontier of a medial axis