Bisections of mass assignments by parallel hyperplanes

TL;DR

This paper develops a new topological approach to prove that certain mass assignments in Euclidean spaces can be bisected by parallel hyperplanes, generalizing previous conjectures and results.

Contribution

It introduces a novel lifting method and computes a new index to extend bisecting results for mass assignments using parallel hyperplanes.

Findings

Proves bisecting of mass assignments by parallel hyperplanes in higher dimensions.

Generalizes the conjecture by Soberón and Takahashi to broader cases.

Establishes conditions involving Stirling numbers for bisecting mass assignments.

Abstract

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.