The topology of surface mediatrices
Given two points of a metric space the mediatrix is the set of all points which have the same distance from both points. In the paper under review the authors examine the topological structure of mediatrices in connected, compact 2-manifolds with respect to the metric induced from a Riemannian metric. Building on previous work [Topology Appl. 153, No. 9, 1421--1433 (2006; Zbl 1093.54007)] they show that all such mediatrices are homeomorphic to a finite simplicial \(1\)-complex. Furthermore, they determine all five possible complexes for the torus and conjecture the set of possible homeomorphism classes of mediatrices for surfaces of higher genus.
scientific article; zbMATH DE number 5080154