Bisections of mass assignments by parallel hyperplanes

TL;DR

This paper introduces a new topological approach to bisecting mass assignments with parallel hyperplanes, generalizing previous conjectures and providing novel computational methods for equivariant fiber bundles.

Contribution

It develops a novel lifting method and computes the parametrized Fadell--Husseini index to extend bisecting results to higher dimensions and more complex mass configurations.

Findings

Any $d+k+m-1$ mass assignments in $ eal^{d+m}$ can be bisected by $k$ parallel hyperplanes under certain conditions.

The new methods generalize all known cases of a conjecture by Soberón and Takahashi.

The approach involves a Borsuk--Ulam-type question on equivariant fiber bundles.

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.