  Equivariant "Ham Sandwich" Theorems are obtained for the finite subgroups G
of the unit spheres S(F) in the classical algebras F = R, C, and H. Given any n
F-valued mass distributions on F^n, it is shown that there exists a
G-equivariant decomposition of F^n into |G| regular convex fundamental regions
which "G-equipartition" each of the n measures, as realized by the vanishing of
the "G-averages" of these regions' measures. Applications for real measures
follow, among them that any n signed mass distributions on R^{(p-1)n} can be
equipartitioned by a single regular p-fan when p a prime number.
