Constant Riesz potentials on a circle in a plane with an application to polarization optimality problems

A characterization for a Riesz s-potential function of a multiset!N of N points in R² is given when s =2-2N and the potential function is constant on a circle in R². The characterization allows us to partially prove a conjecture of Nikolov and Rafailov that if the potential function is constant on a circle - then the points in !N should be equally spaced on a circle concentric to - . As an application of constant Riesz s-potential functions, we also find all maximal and minimal polarization constants and configurations of two concentric circles in R² for certain values of s.