Topology of the GrÜnbaum–hadwiger–ramos hyperplane mass partition problem

Abstract

In 1960 Grünbaum asked whether for any finite mass in ℝd there are d hyperplanes that cut it into 2d 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. More generally, Ramos (1996) asked for the smallest dimension Δ(j, k) in which for any j masses there are k affine hyperplanes that simultaneously cut each of the masses into 2k equal parts. At present the best lower bounds on Δ(j, k) are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica and Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology. We give a critical review of the work on the Grünbaum–Hadwiger–Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that Δ(j, 2) =½(3j +1) in the cases when j − 1 is a power of 2, j ≥ 5.