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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