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 2X ab ∈ H 1 (D 8 ; Z) ∼ = Z/4, where X ab 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 BourginYang theorems, Ergodic Theory Dynam. Systems 8 * (1988) 73-85].

The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes

“…• The product scheme is the classical one, already considered in [27] and [21]. The problem is translated to the problem of the existence of a W k -equivariant map,…”

Hyperplane equipartitions plus constraints

“…As observed in [16,Theorem 4.1] for the proof of ∆(m; k) ≤ U (m; k), Π (α1,...,α k ) =0 (x α1 + · · · + x α k ) arising from the Z ⊕k 2 -representation U k is Dickson and can be expressed explicitly (see, e.g., [21]) as…”

“…In favorable circumstances the vanishing of such maps is guaranteed by Borsuk-Ulam type theorems which rely on the calculation of advanced algebraic invariants such as the ideal-valued index theory of Fadell-Husseini [11] or relative equivariant obstruction theory. Such methods have produced relatively few exact values of ∆(m; k), however, which at present are known for • all m if k = 1 (the well-known Ham Sandwich Theorem ∆(m; 1) = m), • three infinite families if k = 2: ∆(2 q+1 + r; 2) = 3 · 2 q + ⌊3r/2⌋, r = −1, 0, 1 and q ≥ 0 [16,5,6], • three cases if k = 3: ∆(1; 3) = 3 [13], ∆(2; 3) = 5 [5], and ∆(4; 3) = 10 [5], and relies only on Z ⊕k 2 -equivariance rather than the full symmetries of S ± k and was given by Mani-Levitska, Vrećica, andŽivaljević in 2007 [16]. It is easily verified that U (m; k) = L(m; k) follows from (1.3) only when (a) k = 1 or when (b) k = 2 and m = 2 q+1 − 1, with a widening gap between U (m; k) and L(m; k) as r tends to zero, and as either q or r increases.…”

“…In addition to the product scheme primarily used (see, e.g., [6,16,23]) one can also consider as in [5,8] the k-fold join…”

Equipartitions of a mass in boxes

“…The most recent and the most complete answers to this question are given in [4]. The most important open question is whether every mass distribution in R 4 admits an equipartition by 4 hyperplanes in 16 hyperorthants.…”