  In the present paper we develop the theory of minimization for energies with
multivariate kernels, i.e. energies, in which pairwise interactions are
replaced by interactions between triples or, more generally, $n$-tuples of
particles. Such objects, which arise naturally in various fields, present
subtle differences and complications when compared to the classical two-input
case. We introduce appropriate analogues of conditionally positive definite
kernels, establish a series of relevant results in potential theory, explore
rotationally invariant energies on the sphere, and present a variety of
interesting examples, in particular, some optimization problems in
probabilistic geometry which are related to multivariate versions of the Riesz
energies.
