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- W2748278604 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." @default.
- W2748278604 created "2017-08-31" @default.
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- W2748278604 date "2010-11-08" @default.
- W2748278604 modified "2023-09-25" @default.
- W2748278604 title "Mass Partitions via Equivariant Sections of Stiefel Bundles" @default.
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