Centerlines of regions in the sphere

Abstract

Given a region U in the 2-sphere S such that the boundary of U contains at least two points, let D ( U ) be the collection of open circular disks (called maximal disks) in U whose boundary meets the boundary of U in at least two points and let U 2 be the collection of all regions U ⊂ S such that for each D ∈ D ( U ) , D meets the boundary of U in at most two points. In this paper we study geometric properties of regions U ∈ U 2 . We show for such U that the centerline (i.e., the set of centers of maximal disks) is always a smooth, connected 1-manifold. We also show that the boundary of U has at most two components and, if it has exactly two components, then the boundary is locally connected. These results are related the set of points E ( X , Y ) which are equidistant to two disjoint closed sets X and Y. In particular we investigate when the equidistant set is a 1-manifold.