Equipartitions of measures in $\mathbb{R}^4$

Details Equipartitions of measures in $\mathbb{R}^4$ Equipartitions of measures in $\mathbb{R}^4$

2004-12-23

arxiv.org/abs/math/0412483v1

Mathematics

Combinatorics

Scientific paper

We prove that each measure $\mu$ in $R^4$ admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional, affine subspace $L$ of $R^4$. Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on the Koschorke's exact singularity sequence and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4.

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