Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres

Details Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres

2011-09-04

arxiv.org/abs/1109.0721v2

Mathematics

Combinatorics

14 pages. Subsumes 1006.4614 and 1009.5896, including more applications

Scientific paper

Equivariant "Ham Sandwich" Theorems are obtained for the finite subgroups of the unit spheres S^{d-1}, d=1,2,4. Given any F-valued mass distributions on F^n and any non-zero finite subgroup G of the unit sphere S^{d-1} in F= R, C, or H, it is shown that there exists a collection of fundamental G-regions partitioning F^n which "G-Equipartition" each of the n measures, as realized by the simultaneous vanishing of the "G-averages" of the regions' measures. Equipartition results 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 for any prime number p.

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