G-Ham Sandwich Theorems: Balancing measures by finite subgroups of spheres

Equivariant Ham Sandwich Theorems are obtained for the classical algebras F=R,C, and H and the finite subgroups G of their unit spheres. Given any n F-valued Borel measures on Fn and any n-dimensional free F-unitary representation of G, it is shown that there exists a Voronoi partition of Fn naturally determined by G which “G-balances” each measure, as realized by the simultaneous vanishing of each “G-average” of the measures of the partitionʼs isometric fundamental domains. Applications for real measures follow, among them that any n signed mass distributions on C(p−1)n/2 can be equipartitioned by a single complex regular p-fan if p is an odd prime.