Many partitions of mass assignments

TL;DR

This paper develops a comprehensive topological framework for partitioning mass assignments with hyperplanes on Euclidean vector bundles, extending previous results and unifying various theorems in the field.

Contribution

It introduces a new configuration test map scheme and topological methods to generalize and unify existing mass partition results on Euclidean bundles.

Findings

Reproves known mass partition theorems using new methods

Extends results to arbitrary Euclidean vector bundles

Provides a unified topological framework for various partition problems

Abstract

In this paper, extending the recent work of authors with Calles Loperena and Dimitrijevi\'c Blagojevi\'c, we give a general and complete treatment of a problem of partition of mass assignments with prescribed arrangements of hyperplanes on Euclidean vector bundles. Using a new configuration test map scheme, as well as an alternative topological framework, we are able to reprove known results, extend them to arbitrary bundles as well as to put various types of constraints on the solutions. Moreover, the developed topological methods allow us to give new proofs and extend results of Guth and Katz, Schnider, and Sober\'on and Takahashi. In this way we place all these results under one ``roof''.