Equipartitions of a mass in boxes

The aim of this paper is to provide the sufficient condition for a mass distribution in Rd 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. Vrećica, R. Živaljević, 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.