Configurations with Nonuniform Distribution

Abstract

In this chapter we discuss the following generalization of the minimal discrete energy problem. Let \((\subset \mathbb {R})\) be compact metric space and let \(w:A\times A\rightarrow [0,\infty ]\) be a given function which we will call the weight. For a given N-point configuration \(\omega _N=\{x_1,\ldots , x_N\}\) on A and a given number \(s>0\), define the (w, s)-energy of \(\omega _N\) by \( E^w_s(\omega _N):=\sum \limits _{i=1}^{N}\sum \limits _{j=1\atop j\ne i}^{N}\frac{w(x_i, x_j)}{|x_i - x_j|^s}. \)