Potential theory with multivariate kernels

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.

23 pages

References in corpus (6)

• New upper bounds for kissing numbers from semidefinite programming
• The Perfect Glass Paradigm: Disordered Hyperuniform Glasses Down to Absolute Zero
• Face-centered cubic crystallization of atomistic configurations
• $k$-point semidefinite programming bounds for equiangular lines
• Stolarsky's invariance principle for projective spaces
• Stolarsky principle and energy optimization on the sphere