A union of an arrangement of affine hyperplanes \(\mathcal {H}\) in \(\mathbb {R}^d\) is the real algebraic variety associated to the principal ideal generated by the polynomial \(p_{\mathcal {H}}\) given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on \(\mathbb {R}^d\) is bisected by the arrangement of affine hyperplanes \(\mathcal {H}\) if the measure on the “non-negative side” of the arrangement \(\{x\in \mathbb {R}^d:p_{\mathcal {H}}(x)\ge 0\}\) is the same as the measure on the “non-positive” side of the arrangement \(\{x\in \mathbb {R}^d : p_{\mathcal {H}}(x)\le 0\}\). In 2017 Barba, Pilz & Schnider considered special, as well as modified cases of the following measure partition hypothesis: For a given collection of j finite Borel measures on \(\mathbb {R}^d\) 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 \(j=4\). Furthermore, they conjectured that every collection of j measures on \(\mathbb {R}^d\) 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 Blagojević, Frick, Haase & Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grünbaum–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 \(\mathbb {R}^{2^a+\ell }\), where \(1\le \ell \le 2^a-1\), 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 [8].

\(\mathbb {R}^d\) 中的仿射超平面\(\mathcal {H}\) 的并集是与由多项式\(p_{\mathcal { H}}\)作为定义排列超平面的多项式的乘积给出。\(\mathbb {R}^d\)上的有限 Borel 测度被仿射超平面\(\mathcal {H}\)的布置平分，如果在布置\(\ {x\in \mathbb {R}^d:p_{\mathcal {H}}(x)\ge 0\}\)与排列\(\ {x\in \mathbb {R}^d : p_{\mathcal {H}}(x)\le 0\}\). 在 2017 年 Barba，Pilz & Schnider 考虑了以下度量划分假设的特殊情况以及修改情况：对于在\(\mathbb {R}^d\)上的j 个有限 Borel 度量的给定集合，存在k元素仿射将每个度量同时平分为相等的两半的超平面排列。他们表明在\(d=k=2\)和\(j=4\)的情况下存在同时二分。此外，他们推测\(\mathbb {R}^d\)上的每个j测度集合都可以同时用k元素仿射超平面排列二等分，条件是\(d\ge \lceil j/k\rceil \). 该猜想在 2018 年 Hubard 和 Karasev 在\(d\ge j/k=2^a\)的情况下得到证实。 在本文中，我们使用 Blagojević, Frick 的框架对 Hubard 和 Karasev 结果进行了不同的证明, Haase & Ziegler (2016)，基于 tom Dieck 的等变相对阻塞理论，该理论是为处理 Grünbaum-Hadwiger-Ramos 超平面测度划分问题而开发的。此外，这种方法让我们能够证明更多，对于\(2^a(2h+1)+\ell \)对\(\mathbb {R}^{2^a+\ell }\) 的测量的 每个集合，其中\(1\le \ell \le 2^a-1\)，存在一个\((2h+1)\)- 元素仿射超平面排列，同时平分所有这些。我们的结果被扩展到球形排列的情况，并由 Crabb [8] 以一种漂亮的方式通过替代方法进行了谴责。