Hyperplane Mass Partitions via Relative Equivariant Obstruction Theory

The Grünbaum--Hadwiger--Ramos hyperplane mass partition problem was introduced
by Grünbaum (1960) in a special case and in general form by Ramos (1996).
It asks for the «admissible» triples $(d,j,k)$ such that for any $j$
masses in $\R^{d}$ there are $k$ hyperplanes that cut each of the masses
into $2^{k}$ equal parts. Ramos' conjecture is that the Avis--Ramos necessary
lower bound condition $dk\ge j(2^{k}-1)$ is also sufficient. We develop
a «join scheme» for this problem, such that non-existence of an ${\Sym_{k}^\pm}$-equivariant
map between spheres $(S^{d})^{*k} \rightarrow S(W_{k}\oplus U_{k}^{\oplus j})$
that extends a test map on the subspace of $(S^{d})^{*k}$ where the hyperoctahedral
group $\Sym_{k}^\pm$ acts non-freely, implies that $(d,j,k)$ is admissible.
For the sphere $(S^{d})^{*k}$ we obtain a very efficient regular cell decomposition,
whose cells get a combinatorial interpretation with respect to measures
on a modified moment curve. This allows us to apply relative equivariant
obstruction theory successfully, even in the case when the difference
of dimensions of the spheres $(S^{d})^{*k}$ and $S(W_{k}\oplus U_{k}^{\oplus
j})$ is greater than one. The evaluation of obstruction classes leads
to counting problems for concatenated Gray codes. Thus we give a rigorous,
unified treatment of the previously announced cases of the Grünbaum--Hadwiger--Ramos
problem, as well as a number of new cases for Ramos' conjecture.

2010 Mathematics Subject Classification: 55N25, 51N20, 52A35, 55R20

Keywords and Phrases: Hyperplane mass partition problem, equivariant topological combinatorics, equivariant obstruction theory

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