# Optimal measures for $p$-frame energies on spheres

@article{Bilyk2022OptimalMF,
title={Optimal measures for \$p\$-frame energies on spheres},
author={Dmitriy Bilyk and Alexey A. Glazyrin and Ryan Matzke and Josiah Park and O. V. Vlasiuk},
journal={Revista Matem{\'a}tica Iberoamericana},
year={2022}
}
• D. Bilyk, +2 authors O. Vlasiuk
• Published 2 August 2019
• Mathematics
• Revista Matemática Iberoamericana
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised to a positive power $p$. 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…
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• Mathematics, Physics
Journal of Functional Analysis
• 2021
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Constructing high order spherical designs as a union of two of lower order
• Mathematics
• 2019
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Minimization of the probabilistic p-frame potential
• Mathematics
• 2012
We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their
Optimal simplices and codes in projective spaces
• Mathematics
• 2016
We find many tight codes in compact spaces, i.e., optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto
Positive Definite Functions on Spheres
• 1994
In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on
t-Designs in Projective Spaces
In this work, the generalized hexagon on 819 points, with parameters (2, 8), is included as a 5-design in the Cayley Plane OP, meeting the absolute bound.
Geodesic distance Riesz energy on the sphere
• Mathematics
Transactions of the American Mathematical Society
• 2018
We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for
Energy functionals, numerical integration and asymptotic equidistribution on the sphere
• Computer Science, Mathematics
J. Complex.
• 2003
It is deduced that point configurations which are extremal for the Riesz energy are asymptotically equidistributed on Sd for 0 ≤s≤d as N → ∞ and explicit rates of convergence for the special case s = d, which had been open are presented.