  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.
