Riesz kernel energy

Summary

The Riesz $s$-energy of a positive measure $\mu$ on $\R^d$ is the double integral of the kernel $\abs{x - y}^{-s}$ against $\mu \times \mu$. For $s = d - 2$ with $d \ge 3$ it is the Newtonian energy of a measure, and for other $s$ it is the energy attached to the Riesz potential. Minimising this energy over probability measures on a compact set defines the Riesz capacity of the set. The discrete version, a sum over pairs of points, is the energy in problems on optimal point configurations such as the Thomson problem.