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

TL;DR

This paper explores how finite metric spaces like polygons and cycle graphs can be distinguished using invariants such as the distance multiset and magnitude, revealing their effectiveness and limitations.

Contribution

It constructs examples of non-congruent spaces with identical invariants and analyzes the conditions under which regular polygons are uniquely identified by these invariants.

Findings

Regular polygons are determined by the distance multiset among planar metric spaces.

Explicit families of homometric but non-congruent circular metric spaces are constructed.

The paper identifies cases where regular polygons and cycle graphs are distinguished by 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.