Shortest arcs in closed planar disks vary continuously with the boundary

Given a triple (D, a, b) where D a closed topological disk in the plane with distinct points {a, b} subset of D there is not necessarily a path alpha of finite length connecting a to b in D. Nevertheless the notion of shortest are extends naturally to the topological category as follows. There exists a continuous operator geo which assigns to each triple (D, a, b) a closed are alpha subset of D such that alpha connects a and b, and moreover alpha has uniquely minimal finite Euclidean arclength whenever possible. (C) 1999 Elsevier Science B.V. All rights reserved.