Hyperplane mass partitions via relative equivariant obstruction theory

The Gr\"unbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Gr\"unbaum (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 $G_k$-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 $G_k$ 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 Gr\"unbaum-Hadwiger-Ramos problem, as well as a number of new cases for Ramos' conjecture.