Mass Partitions via Equivariant Sections of Stiefel Bundles

Abstract

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of $m$ mass distributions on ${\mathbb{R}}^n$, the existence of $k$ affinely independent regular $q$-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a ${\mathbb{Z}}_q$-equivariant section of the Stiefel bundle $V_k({\mathbb{F}}^n)$ over $S({\mathbb{F}}^n)$, where $V_k({\mathbb{F}}^n)$ is the Stiefel manifold of all orthonormal $k$-frames in ${\mathbb{F}}^n,\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and $S({\mathbb{F}}^n)$ is the corresponding unit sphere. For example, the parallelizability of $\mathbb{R}{P}^n$ when $n=2,4$, or $8$ implies that any two masses on ${\mathbb{R}}^n$ can be simultaneously bisected by each of $(n-1)$ pairwise-orthogonal hyperplanes, while when $q=3$ or 4, the triviality of the circle bundle $V_2({\mathbb{C}}^2)/{\mathbb{Z}}_q$ over the standard Lens Spaces $L^3(q)$ yields that for any mass on ${\mathbb{R}}^4$, there exist a pair of complex orthogonal regular $q$-fans, each of which equipartitions the mass.

arXiv comment11 pages, final version

References in corpus (1)

• Extensions of theorems of Rattray and Makeev

Cited by in corpus (1)

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