Universal minima of discrete potentials for sharp spherical codes

TL;DR

This paper establishes universal lower bounds for discrete potentials on spheres related to sharp codes, extending to various spherical designs and special configurations like the icosahedron, $E_8$, and the Leech lattice, with implications for potential minimization.

Contribution

It introduces universal bounds for potentials on sharp spherical codes and extends these results to T-designs, special configurations, and specific potential functions.

Findings

Sharp codes attain universal lower bounds for polarization.

Universal bounds apply to T-designs and configurations like the icosahedron and Leech lattice.

Certain potential functions lead to minima at code points, confirming optimality of sharp codes.

Abstract

This article is devoted to the study of discrete potentials on the sphere in $\mathbb{R}^n$ for sharp codes. We show that the potentials of most of the known sharp codes attain the universal lower bounds for polarization for spherical $\tau$-designs previously derived by the authors, where ``universal'' is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products. We also extend our universal bounds to $T$-designs and the associated polynomial subspaces determined by the vanishing moments of spherical configurations and thus obtain the minima for the icosahedron, dodecahedron, and sharp codes coming from $E_8$ and the Leech lattice. For this purpose, we investigate quadrature formulas for certain subspaces of Gegenbauer polynomials $P^{(n)}_j$ which we call PULB subspaces, particularly those having basis $\{P_j^{(n)}\}_{j=0}^{2k+2}\setminus \{P_{2k}^{(n)}\}.$ Furthermore, for potentials with $h^{(\tau+1)}<0$ we prove that the strong sharp codes and the antipodal sharp codes attain the universal bounds and their minima occur at points of the codes. The same phenomenon is established for the $600$-cell when the potential $h$ satisfies $h^{(i)}\geq 0$, $i=1,\dots,15$, and $h^{(16)}\leq 0.$