Discrete Riesz energy uniquely identifies regular n-gons in the plane

Establish that for every integer n > 3, among all n-point configurations in the Euclidean plane R^2, the discrete Riesz energy function B_X(z) uniquely determines the regular n-gon R_n up to isometry. This extends the proven identification result, which holds when n is not a multiple of 3 and n > 30, to all n.

Background

The paper shows that, in general metric spaces, the discrete Riesz energy function B_X(z) exactly encodes the multiset of interpoint distances and therefore cannot distinguish non-isometric finite metric spaces (mutants) sharing the same distance multiset. Consequently, neither cycle graphs nor regular n-gons can be identified by B_X(z) without restricting the ambient class of spaces.

Restricting to n-point subsets of the plane, the authors prove (Theorem 4.4) that the discrete Riesz energy identifies the regular n-gon for all integers n that are not multiples of 3 and satisfy n > 30. They explicitly conjecture that this identification property holds for any n, leading to the open problem of removing the current technical restriction on n.