Energy on Spheres and Discreteness of Minimizing Measures

Ryan William Matzke, Dmitriy Bilyk, Oleksandr Vlasiuk, Alexey Glazyrin, Josiah Park

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Article number108995
JournalJournal of Functional Analysis
Issue number11
Publication statusPublished - 1 Jun 2021


  • Energy Minimization
  • Spherical Codes
  • Spherical designs


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