Optimal measures for pp-frame energies on spheres

Dmitriy Bilyk ^[1] ; Alexey Glazyrin ^[4] ; Ryan Matzke ^[1] ; Josiah Park ^[2] ; Oleksandr Vlasiuk ^[3]
1. [1] University of Minnesota
  
  University of Minnesota
  
  City of Minneapolis, Estados Unidos
2. [2] Georgia Institute of Technology
  
  Georgia Institute of Technology
  
  Estados Unidos
3. [3] Florida State University
  
  Florida State University
  
  Estados Unidos
4. [4] University of Texas Rio Grande
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Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 4, 2022, págs. 1129-1160
Idioma: inglés
DOI: 10.4171/RMI/1329
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Resumen
- We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the pp-frame energies, i.e., energies with the kernel given by the absolute value of the inner product raised to a positive power pp. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the 600-cell for several ranges of pp in different dimensions. Our methods apply to a much broader class of potential functions, namely, those which are absolutely monotonic up to a particular order.