Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres

TL;DR

This paper extends the classical ham sandwich theorem to finite subgroups of spheres in real, complex, and quaternionic spaces, establishing G-equivariant equipartitions for multiple measures with applications to real measures and prime p-fans.

Contribution

It introduces G-equivariant ham sandwich theorems for finite subgroups of spheres, providing new equipartition results in algebraic and geometric settings.

Findings

Existence of G-equivariant decompositions for measures in F^n

G-equipartition of measures via regular convex fundamental regions

Application to equipartitioning signed measures with p-fans for prime p

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 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.