Multiple farthest points on Alexandrov surfaces

Abstract

Abstract The farthest point mapping on compact surfaces, associating to each point x of the surface the set of absolute maxima of the intrinsic distance from x , is for some surfaces single-valued and a homeomorphism, while for other surfaces it is not single-valued, and not surjective. These two big classes are not very well understood. For instance it is still unknown whether, say in the convex case, the second class is dense. For a C 2 metric on both surfaces and the space of surfaces, the first class has, however, nonempty interior. We describe various properties of the sets of critical points, and of relative and absolute maxima of distance functions, and find several connections between them. We see for example that, on smooth surfaces homeomorphic to S 2 , a point cannot be critical with respect to more than one other point. Sufficient conditions for a surface to belong to the second class will be formulated and a particular Tannery surface belonging to the boundary of both classes will be presented.