Hyperplane mass partitions via relative equivariant obstruction theory

Abstract

The Grunbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grunbaum (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 Rd there are k hyperplanes that cut each of the masses into 2k equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition dk ≥ j(2k-1) is also sufficient. We develop a "join scheme" for this problem, such that non-existence of an S± k-equivariant map between spheres (Sd)*k S that extends a test map on the subspace of (Sd)*k where the hyperoctahedral group S± k acts non-freely, implies that (d, j, k) is admissible. For the sphere (Sd)*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 (Sd)*k and 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 Grunbaum-Hadwiger-Ramos problem, as well as a number of new cases for Ramos' conjecture.