Hyperplane mass partitions via relative equivariant obstruction theory

Abstract

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 ${\mathbb{R}}^d$ 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 ${\mathfrak{S}}_k^{\pm }$-equivariant map between spheres $(S^d{)}^{\ast k}\to S(W_k\oplus U_k^{\oplus j})$ that extends a test map on the subspace of $(S^d{)}^{\ast k}$ where the hyperoctahedral group ${\mathfrak{S}}_k^{\pm }$ acts non-freely, implies that $(d,j,k)$ is admissible. For the sphere $(S^d{)}^{\ast k}$ 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 $(S^d{)}^{\ast k}$ 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.