Equidistant Sets in Spaces of Bounded Curvature

Given a metric space (X,d), and two nonempty subsets A,B ⊆ X, we study the properties of the set of points of equal distance to A and B, which we call the equidistant set E(A,B). In general, the structure of the equidistant set is quite unpredictable, so we look for conditions on the ambient space, as well as the given subsets, which lead to some regularity of the properties of the equidistant set. At a minimum, we will always require that X is path connected (so that E(A,B) is nonempty) and A and B are closed and disjoint (trivially, Ā ∩ B ⊂ E(A,B)). Historically, the equidistant set has primarily been studied with the assumptions: (i) X is Euclidean space and A, B are closed and disjoint; or (ii) X is a compact smooth surface and A, B are singleton sets. We combine and extend on these requirements by examining equidistant sets with the conditions that X is a compact Alexandrov surface (of curvature bounded below) and A,B are compact and disjoint. Significantly, we find E(A,B) is always a finite simplicial 1-complex. The techniques developed are also applied to answer two open problems concerning equidistant sets in the Euclidian plane. In particular, we show that if A and B are disjoint closed subsets of the plane, then E(A,B) is a topological 1-manifold, and the Hausdorff dimension of E(A,B) is 1