More bisections by hyperplane arrangements

Abstract

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojevi\'c, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Gr\"unbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.