Identification of finite circular metric spaces by magnitude and Riesz energy

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.