Minimizing energy on locally compact spaces: existence and approximation

In this paper we study the existence and approximation of solutions of the energy minimization problem on general convex sets of positive Radon measures by using direct methods. To establish the existence of solutions we mainly use the vague topology and weak hypotheses on the kernel, for example one of them is the existence of vague cluster points to Cauchy sequences. Furthermore, we give sufficient conditions on the convex set to assure the existence of minimizing sequences of discrete measures. This can be thought of as a generalization of the Transfinite Diameter method and of the Frotsman techniques. In addition, when the kernel is extended continuously the computation of the minimal energy or of minimizing sequences is operative. Finally, we obtain regular capacities on the class of the positive Radon measures, for which every subset is capacitable