Topology of the Grünbaum–Hadwiger–Ramos hyperplane mass partition problem

Abstract

In 1960 Grünbaum asked whether for any finite mass in R d there are d hyperplanes that cut it into 2 d equal parts. This was proved by Hadwiger (1966) for d ≤ 3, but disproved by Avis (1984) for d ≥ 5, while the case d = 4 remained open.

Hyperplane equipartitions plus constraints

“…Following a detailed review of the CS/TM paradigm in Section 3 as previously applied to Question 1, we discuss in Section 4 its modifications to our constrained cases, the heart of which (as in [4]) lies in the imposition of further conditions on the target space. In Section 5, we show how a reduction trick (Lemma 5.1) from [6] immediately implies that upper bounds ∆(m + 1; k) ≤ d + 1 obtained via that scheme produce upper bounds ∆ ⊥ (m; k) ≤ d for full orthogonality (Theorem 5), an observation we owe to Florian Frick. This restriction technique does not produce cascades or specified affine containment, however, and answers to Question 2 under a variety of these constraints, including partial or full orthogonality, are given in Section 6 using cohomological methods (Theorems 6.2 and 6.4, Propositions 6.5-6.6).…”

“…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…”

“…Nonetheless, in the following two sections we show that ∆ ⊥ ((2 q+2 − 2, 1); 2) = 3 ·2 q+1 − 2 (Corollary 7.1), and in Theorem 6.4 we recover ∆ ⊥ (m; k) ≤ U (m; k) above while including cascades and affine containment constraints which remove all remaining degrees of freedom in (2.1). 6 Optimizing the Topological Upper Bound U (m; k)…”

Ham-Sandwich Cuts and Center Transversals in Subspaces

“…1 for an example. A similar setting, where the directions of the hyperplanes are somewhat restricted, has been studied by several authors [1,5,13]. We say that L bisects a mass distribution µ if µ(R + ) = µ(R − ).…”