Universal minima of potentials of certain spherical designs contained in the fewest parallel hyperplanes

TL;DR

This paper identifies all universal minimum points for specific spherical configurations, extending previous results and analyzing properties of spherical designs in minimal hyperplane arrangements.

Contribution

It determines universal minima for several high-dimensional spherical codes and extends known results to new configurations and symmetrizations.

Findings

Universal minima for 16-point sharp code on S^4

Universal minima for demihypercube on S^d, d≥5

Properties of spherical designs in minimal hyperplanes

Abstract

We find the set of all universal minimum points of the potential of the $16$-point sharp code on $S^4$ and (more generally) of the demihypercube on $S^d$, $d\geq 5$, as well as of the $2_{41}$ polytope on $S^7$. We also extend known results on universal minima of three sharp configurations on $S^{20}$ and $S^{21}$ containing no antipodal pair to their symmetrizations about the origin. Finally, we prove certain general properties of spherical $(2m-1)$-designs contained in as few as $m$ parallel hyperplanes (all but one configuration considered here possess this property).