  We investigate the nature of subsets of spheres which satisfy a tameness
condition associated with the Bieri-Groves conjecture on cohomological
finiteness conditions for metabelian groups. We find that there is a natural
polyhedrality in a crucial special case. In the case of the two dimensional
sphere we establish a strong polyhedrality condition for certain open sets
which are maximal subject to satisfying the tameness condition that subsets of
three or fewer points are contained in open hemispheres. Many examples are
included.

Group Theory;Combinatorics;Metric Geometry