Potential theory with multivariate kernels

Dmitriy Bilyk, Damir Ferizović, Alexey Glazyrin, Ryan W Matzke, Josiah Park, Oleksandr Vlasiuk

Research output: Contribution to journalArticlepeer-review

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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.

Original languageEnglish (US)
Pages (from-to)2907-2935
Number of pages29
JournalMathematische Zeitschrift
Issue number3
StatePublished - Jul 2022

Bibliographical note

Funding Information:
The authors thankfully acknowledge the support of this research: NSF grant DMS 1665007 and 2054606, the Simons Foundation collaboration grant for mathematicians 712810 (D. Bilyk), the Austrian Science Fund (FWF): F5503 “Quasi-Monte Carlo Methods” and FWF: W1230 “Doctoral School Discrete Mathematics”, and the Austrian Marshall Plan Foundation (D. Ferizović), NSF grant DMS 2054536 (A. Glazyrin), the NSF Graduate Fellowship 00039202 and the UMN Doctoral Dissertation Fellowship (R. Matzke), AMS-Simons Travel Grant and Postdoctoral Travel Award from the FSU Office of Postdoctoral Affairs (O. Vlasiuk), NSF grants CCF 1934904 and DMS 1600693 (J. Park)

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.


  • Optimal measures
  • Positive definite kernels
  • Potential energy minimization


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