SHORTEST CURVES IN JORDAN REGIONS VARY CONTINUOUSLY WITH THE BOUNDARY

Suppose a closed loop of wire in the plane defines a Jordan region, and two points in the interior of that region are joined by a taut rubber band constrained to lie in the region by that wire boundary. Now suppose that both the wire and the interior points are continuously perturbed (so that the perturbed endpoints lie in the interior of the perturbed Jordan region at each stage). Then the ruber band moves continuously. We provide a mathematical formulation and proof of this assertion. (C) 1994 Academic Press, Inc.