More bisections by hyperplane arrangements

TL;DR

This paper presents a new proof for a measure partition problem using equivariant obstruction theory, extending previous results to more measures and arrangements, and includes applications to spherical arrangements.

Contribution

It offers a different proof of the Hubard and Karasev result and extends the measure bisecting arrangements to more complex cases with additional measures.

Findings

Established existence of hyperplane arrangements bisecting multiple measures in higher dimensions.

Extended measure bisecting results to arrangements with more measures and hyperplanes.

Provided alternative proofs and applications to spherical 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.