标题： 质量分配的平行超平面二分法 显示英文标题

标题： Bisections of mass assignments by parallel hyperplanes

摘要： 在本文中，我们证明了关于欧几里得向量丛上平行超平面对质量分配的二分结果。 我们的方法包括开发一种新颖的提升方法来定义配置空间-测试映射方案，这将问题转化为等变纤维丛上的Borsuk-Ulam型问题，同时还包括对参数化Fadell-Husseini指标的新计算。 作为主要应用，我们表明，任何$d+k+m-1$质量分配到$\mathbb{R}^{d+m}$中的线性$d$空间，可以在至少一个$d$空间中被$k$个平行超平面二分，前提是第二类斯特林数$S(d+k+m-1, k)$为奇数。 这推广了Soberón和Takahashi的一个猜想的所有已知情况，该猜想断言，任何$d+k-1$个度量在$\mathbb{R}^d$中可以通过$k$个平行超平面进行二分。 显示英文摘要

摘要： In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any $d+k+m-1$ mass assignments to linear $d$-spaces in $\mathbb{R}^{d+m}$ can be bisected by $k$ parallel hyperplanes in at least one $d$-space, provided that the Stirling number of the second kind $S(d+k+m-1, k)$ is odd. This generalizes all known cases of a conjecture by Sober\'on and Takahashi, which asserts that any $d+k-1$ measures in $\mathbb{R}^d$ can be bisected by $k$ parallel hyperplanes.