Optimal Measures for Multivariate Geometric Potentials

We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of points, which serve as multi-input generalizations of the well-known Riesz potentials for pairwise interaction. One of such potentials is volume squared of the simplex with vertices at the $k \ge 3$ given points: we show that the arising energy is maximized by balanced isotropic measures, in contrast to the classical two-input energy. These results are used to obtain interesting geometric optimality properties of the regular simplex. As the main machinery, we adapt the semidefinite programming method to this context and establish relevant versions of the $k$-point bounds.