Constant Generalized Riesz Potential Functions and Polarization Optimality Problems

Nattapong Bosuwan

Abstract

An extension of a conjecture of Nikolov and Rafailov [N. Nikolov, R. Rafailov, On extremums of sums of powered distances to a finite set of points, Geom. Dedicata 167(1) (2013) 69--89] by considering the following  potential function defined on $\mathbb{R}^2$:

$$f_s(x)=\sum_{j=1}^N \left(|x-x_j|^2+h\right)^{-s/2},\quad \quad h\geq0,$$

for $s=2-2N$ is given.  We obtain a characterization of sets of $N$ distinct points $\{x_1,x_2,\ldots,x_N\}$ such that $f_{2-2N}$ is constant on some circle in $\mathbb{R}^2.$  Using this characterization, we prove some special cases of this new conjecture. The other problems considered in this paper are polarization optimality problems. We find all maximal and minimal polarization constants and configurations of two concentric circles in $\mathbb{R}^2$ using the above potential function for certain values of $s.$