Counting Gray codes for an improved upper bound of the Gr\"unbaum-Hadwiger-Ramos problem

Abstract

We give an improved upper bound for the Gr\"unbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that partition it into $2^k$ sets of equal size with respect to all measures. This is an improvement to the previous bound $d \geq 2^{n + k}$ by Mani-Levitska, Vre\'cica & \v{Z}ivaljevi\'c in 2006. This is achieved by classifying the number of certain Gray code patterns modulo 2. The reduction was developed by Blagojevi\'c, Frick, Haase & Ziegler in 2016. It utilizes the group action of the symmetric group $(\mathbb{Z}/2)^k \rtimes \mathfrak{S}_k$ of $k$ oriented hyperplanes. If we restrict to the subgroup $(\mathbb{Z}/2)^k$ as Mani-Levitska et al. we retrieve their bound.