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 $y\in S$. 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 $y$.