  Let $L$ be a set of $n$ axis-parallel lines in $\mathbb{R}^3$. We are are
interested in partitions of $\mathbb{R}^3$ by a set $H$ of three planes such
that each open cell in the arrangement $\mathcal{A}(H)$ is intersected by as
few lines from $L$ as possible. We study such partitions in three settings,
depending on the type of splitting planes that we allow. We obtain the
following results.
  $\bullet$ There are sets $L$ of $n$ axis-parallel lines such that, for any
set $H$ of three splitting planes, there is an open cell in $\mathcal{A}(H)$
that intersects at least~$\lfloor n/3 \rfloor-1 \approx \frac{1}{3}n$ lines.
  $\bullet$ If we require the splitting planes to be axis-parallel, then there
are sets $L$ of $n$ axis-parallel lines such that, for any set $H$ of three
splitting planes, there is an open cell in $\mathcal{A}(H)$ that intersects at
least $\frac{3}{2}\lfloor n/4 \rfloor -1 \approx \left(
\frac{1}{3}+\frac{1}{24}\right) n$ lines. Furthermore, for any set $L$ of $n$
axis-parallel lines, there exists a set $H$ of three axis-parallel splitting
planes such that each open cell in $\mathcal{A}(H)$ intersects at most
$\frac{7}{18} n = \left( \frac{1}{3}+\frac{1}{18}\right) n$ lines.
  $\bullet$ For any set $L$ of $n$ axis-parallel lines, there exists a set $H$
of three axis-parallel and mutually orthogonal splitting planes, such that each
open cell in $\mathcal{A}(H)$ intersects at most $\lceil \frac{5}{12} n \rceil
\approx \left( \frac{1}{3}+\frac{1}{12}\right) n$ lines.
