Identification of finite circular metric spaces by magnitude and Riesz energy

TL;DR

The paper investigates whether finite metric spaces, specifically regular polygons $R_n$ and cycle graphs $C_n$, can be identified by magnitude and the discrete Riesz energy. It introduces isomers—non-isometric spaces sharing the same distance multiset and invariants like the formal magnitude $m_X(q)$ and the energy $B_X(z)$—and analyzes how magnitude, magnitude homology, and energy interact with symmetry. The authors establish precise identifiability results for small $n$ (e.g., $R_n$ identifiable for $n\in\{3,4,5,7\}$ and $C_n$ for $n\in\{3,4,5\}$) and demonstrate broad non-identifiability via isomers for larger $n$, while showing that magnitude homology can distinguish certain isomers even when the magnitude itself cannot. For the discrete Riesz energy, they prove non-identifiability in general but show identifiability when restricting to planar subsets for many $n$ not divisible by $3$, with explicit exceptional configurations for $n$ multiples of $3$ and concrete cases such as $n=9$ and $n=12$.

Abstract

We study the problem whether regular polygons and cycle graphs can be identified by the magnitude or the discrete Riesz energy. We give an affirmative answer for a finite number of cases when the number of vertices is small and a negative answer for an infinite number of wide cases by giving a way to construct non-isometric spaces, which we call isomers, with the same magnitude or the discrete Riesz energy. We study whether isomers can be distinguished by the magnitude homology groups.

Figures (13)

• Figure 1: $n/2$ is even
• Figure 2: $n/2$ is odd
• Figure 3: All the possible configuration of five points and five edges with a cycle (cycles)
• Figure 4: Length $1$ edges
• Figure 6: The solid lines represent edges of length $1$ and the dashed $\sqrt2$.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6
• proof