Topology and combinatorics of partitions of masses by hyperplanes

An old problem in combinatorial geometry is to determine when one or more measurable sets in Rd admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257–1261]. A related topological problem is the question of (non)existence of a map , equivariant with respect to the Weyl group Wk=Bk:=(Z/2)⊕k⋊Sk, where U is a representation of Wk and S(U)⊂U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R8 [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147–167]. The obstruction in this case is identified as the element 2Xab∈H1(D8;Z)≅Z/4, where Xab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel–Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk–Ulam and Bourgin–Yang theorems, Ergodic Theory Dynam. Systems 8* (1988) 73–85].