Antipodal trees and mutually critical points on surfaces

Abstract To any point on an Alexandrov surface homeomorphic to the sphere one can associate a minimal subtree of the cut locus containing all farthest points. It is called the antipodal tree. Two points of a compact orientable Alexandrov surface are called mutually critical if each of them is critical with respect to the other. All points which are mutually critical with a given point form a set. In this paper we show that this set, as well as the set of endpoints of any antipodal tree, are finite.