In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.
Bibliographical noteFunding Information:
We express our gratitude to the following organizations that hosted subsets of the authors during the work on this paper: AIM, ICERM, INI, CIEM, Georgia Tech. The first author was supported by the grant DMS-1665007, the third author was supported by the Graduate Fellowship 00039202, and the fourth author was supported in part by the grant DMS-1600693, all from the US National Science Foundation. This paper is based upon work supported by the National Science Foundation grant DMS-1439786 while D.B. A.G. and O.V. were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the ?Point Configurations in Geometry, Physics and Computer Science? program. This work is also supported in part by EPSRC grant No. EP/K032208/1. We thank Henry Cohn, David de Laat, Axel Flinth, Michael Lacey, Svitlana Mayboroda, Bruno Poggi, Alexander Reznikov, Daniela Schiefeneder, Alexander Shapiro, and Yao Yao for helpful discussions.
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