  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).
