Topology of the Gr\"unbaum-Hadwiger-Ramos hyperplane mass partition problem

Abstract

In 1960 Gr\"unbaum asked whether for any finite mass in $\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le3$, but disproved by Avis (1984) for $d\ge5$, while the case $d=4$ remained open. More generally, Ramos (1996) asked for the smallest dimension $\Delta(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $\Delta(j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vre\'cica \& \v{Z}ivaljevi\'c (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 Gr\"unbaum--Hadwiger--Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that $\Delta(j,2)= \frac12(3j+1)$ in the cases when $j-1$ is a power of $2$, $j\ge5$.