2022/05/18 by Fox, Logan S., Veerman, J. J. P. · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2205.09155
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.