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
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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.

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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