Topology and combinatorics of partitions of masses by hyperplanes

Abstract

An old problem in combinatorial geometry is to determine when one or more measurable sets in R d 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 f : ( S d ) k → S ( U ) , equivariant with respect to the Weyl group W k = B k : = ( Z / 2 ) ⊕ k ⋊ S k , where U is a representation of W k 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 R 8 [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 2 X a b ∈ H 1 ( D 8 ; Z ) ≅ Z / 4 , where X a b 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].