Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres
Abstract
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 Fn, it is shown that there exists a G-equivariant decomposition of Fn 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.
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