The aim of this Paper is to provide the sufficient condition for a mass distribution in R-d to admit an equipartition with a collection of hyperplanes some of which are parallel. The results extend the previously obtained results for the equipartitions with non-parallel hyperplanes. (See [P. Mani-Levitska, S. Vrecica. R. Zivaljevic, Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006) 266-296; E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167].) The paper also serves as the illustration of the applicability and the power of the methods of equivariant topology (more precisely, equivariant index theory) in the problems of geometric combinatorics. (C) 2008 Elsevier Inc. All rights reserved.
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