An \emph{eight-partition} of a finite set of points (respectively, of a continuous mass distribution) in $\mathbb{R}^3$ consists of three planes that divide the space into $8$ octants, such that each open octant contains at most $1/8$ of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $\mathbb{R}^3$ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in $\mathbb{R}^3$ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of $n$ points in~$\mathbb{R}^3$ (with prescribed normal direction of one of the planes) in time $O^{*}(n^{5/2})$.

翻译：有限点集（或连续质量分布）在 $\mathbb{R}^3$ 中的 \emph{八等分} 由三个平面组成，这些平面将空间划分为 $8$ 个卦限，使得每个开卦限包含至多 $1/8$ 的点（或质量）。1966年，Hadwiger 证明了 $\mathbb{R}^3$ 中的任意质量分布均存在一个八等分；此外，可以预先指定三个平面中一个平面的法线方向。有限点集的类似结论可通过标准极限论证得到。我们证明了该结论的以下变体：$\mathbb{R}^3$ 中的任意质量分布（或点集）均存在一个八等分，其中两个平面的交线具有预先指定的方向。此外，我们提出了一种高效算法，可在 $O^{*}(n^{5/2})$ 时间内计算出 $\mathbb{R}^3$ 中 $n$ 个点集的八等分（并指定其中一个平面的法线方向）。