A SOLUTION TO THE ENERGY MINIMIZATION PROBLEM CONSTRAINED BY A DENSITY FUNCTION

We present a new solution to the problem of determining an energy integral which has a unique minimum at a given Borel probability measure on a compact metric space. For a continuous kernel, we show that there exists a unique weight function such that the given measure is an equilibrium measure with respect to the kernel multiplied by the weight function. The weight function is determined as a unique fixed point of a functional operator. Moreover, if the kernel satisfies the energy principle on the space, then the given measure achieves a unique minimum of the energy integral with respect to the weighted kernel. In order to obtain a kernel satisfying the energy principle on Euclidean subspaces, we improve the condition shown by Gneiting for a defining function of a kernel to belong to the Mittal-Berman-Gneiting class. By using the obtained condition, we show related results for the energy with the kernel. Finally, we present practical examples of distributing a finite number of points that are constrained by a density function.