Constant Riesz potentials on a circle in a plane with an application to polarization optimality problems
Nattapong Bosuwana,b, Pornrat Ruengrotc,*
ABSTRACT: A characterization for a Riesz s-potential function of a multiset ωN 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 Γ then the points in ωN should be equally spaced on a circle concentric to Γ. 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.