Topology and Combinatorics of Partitions of Masses by Hyperplanes

Details Topology and Combinatorics of Partitions of Masses by Hyperplanes Topology and Combinatorics of Partitions of Masses by Hyperplanes

2003-10-23

arxiv.org/abs/math/0310377v1

Mathematics

Combinatorics

27 pages, 7 figures

Scientific paper

One of our result is that 5 measurable sets in $R^8$ always admit an equipartition by 2 hyperplanes. This is an instance of a general equipartition problem (formulated by B. Gr{\" u}nbaum and H. Hadwiger) which can be reduced to the question of (non)existence of a $W_k$-equivariant map where $W_k$ is the group of symmetries of a $k$-cube. We show that the computation of relevant cohomology/bordism obstruction classes often reduces to the question of enumerating the classes of immersed curves in $\mathbb{R}^2$ with a prescribed type and number of intersections with the coordinate axes, which in turn leads to a problem of enumerating classes of cyclic signed $AB$-words.

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