Common maxima of distance functions on orientable Alexandrov surfaces

Abstract

We find properties of the sets M_y⁻¹ of all points on a compact orientable Alexandrov surface S, the distance functions of which have a common maximum at y∈S. For example, the components of M_y⁻¹ are arcwise connected and their number is at most max{1,10g−5}, where g is the genus of S. A special attention receives the case of local tree components of M_y⁻¹, providing a relationship to the unit tangent cone at y.