Potential theory with multivariate kernels

TL;DR

This paper extends potential theory to multivariate kernels involving interactions among n-tuples of particles, exploring theoretical foundations, invariance properties, and applications in probabilistic geometry.

Contribution

It introduces analogues of positive definite kernels for multivariate interactions and develops a comprehensive potential theory framework for these complex energies.

Findings

Established conditions for multivariate kernel energies

Analyzed rotational invariance on the sphere

Connected multivariate energies to probabilistic geometry problems

Abstract

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.