Equidistant sets on Alexandrov surfaces

TL;DR

This paper studies the properties of equidistant sets on Alexandrov surfaces, generalizing known results from Riemannian manifolds, and applies findings to address an open question about their Hausdorff dimension.

Contribution

It extends the understanding of equidistant sets to Alexandrov surfaces and characterizes their structure as finite simplicial 1-complexes, also addressing an open problem in Euclidean plane geometry.

Findings

Equidistant sets are finite simplicial 1-complexes on Alexandrov surfaces.

Generalization of known results from Riemannian 2-manifolds.

Provides an answer to an open question on Hausdorff dimension of equidistant sets in the Euclidean plane.

Abstract

We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.