Hyperplane mass partitions via relative equivariant obstruction theory

  • Pavle V.M. Blagojević

  • Florian Frick

  • Albert Haase

  • Günter M. Ziegler

    1001 Beograd Ithaca Serbia NY 14853 and USA Institute of Mathematics
Hyperplane mass partitions via relative equivariant obstruction theory cover
Download PDF

This article is published open access.

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)(d,j,k) such that for any jj masses in R\R^d there are kk hyperplanes that cut each of the masses into 2^k equal parts. Ramos' conjecture is that the Avis--Ramos necessary lower bound condition dkj(2k1)dk\ge j(2^{k}-1) is also sufficient. We develop a «join scheme» for this problem, such that non-existence of an \Symk±{\Sym_{k}^\pm}-equivariant map between spheres (Sd)kS(WkUkj)(S^{d})^{*k} \rightarrow S(W_{k}\oplus U_{k}^{\oplus j}) that extends a test map on the subspace of (Sd)(S^{d})^*k where the hyperoctahedral group \Symk±\Sym_{k}^\pm acts non-freely, implies that (d,j,k)(d,j,k) is admissible. For the sphere (Sd)(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 (Sd)(S^{d})^*k and S(WkUkj)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.

Cite this article

Pavle V.M. Blagojević, Florian Frick, Albert Haase, Günter M. Ziegler, Hyperplane mass partitions via relative equivariant obstruction theory. Doc. Math. 21 (2016), pp. 735–771

DOI 10.4171/DM/544