Centerlines of regions in the sphere

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 U2 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∈U2. 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.