Distinguishing regular polygons, cycle graphs, and circular metric spaces by the distance multiset and magnitude

Abstract

We investigate how effectively finite metric spaces can be distinguished by distance-based invariants. As model spaces, we consider regular polygons, cycle graphs, and their generalization, circular metric spaces, and as invariants we consider the distance multiset, magnitude, and magnitude homology. We construct explicit families of homometric but non-congruent circular metric spaces, and in many even cases these examples also have the same magnitude as the original space. We prove that regular polygons are determined by the distance multiset among planar metric spaces, but not in general. We also determine, for several values of $n$, whether regular $n$-gons and $n$-cycle graphs are determined by magnitude.