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

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