  For large $q$, does the (discrete) uniform distribution on the set of $q!$
permutations of the vector $(1,2,\dots,q)$ closely approximate the (continuous)
uniform distribution on the $(q-2)$-sphere that contains them? These
permutations comprise the vertices of the regular permutohedron, a
$(q-1)$-dimensional convex polyhedron. Surprisingly to me, the answer is
emphatically no: these permutations are confined to a negligible portion of the
sphere, and the regular permutohedron occupies a negligible portion of the
ball. However, $(1,2,\dots,q)$ is not the most favorable configuration for
approximate spherical uniformity of permutations. Unlike the permutations of
$(1,2,\dots,q)$, the normalized surface area of the largest empty spherical cap
among the permutations of the most favorable configuration approaches 0 as
$q\to\infty$. Several open questions are posed.
