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

TL;DR

This paper reviews the topology-based approaches to the hyperplane mass partition problem, identifies gaps in previous proofs, and establishes exact values of the minimal dimension for certain cases.

Contribution

It critically analyzes prior work on the problem and proves the exact value of elta(j,2)or specific values of j where j-1 is a power of two.

Findings

Identified gaps in previous proofs of the problem.

Established elta(j,2)= (3j+1)/2or j-1 being a power of two.

Provided a comprehensive review of the problem's topological methods.

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 $2$, $j\ge5$.