Partitions of mass assignments with spheres and wedges

TL;DR

This paper extends classic mass partition theorems to mass assignments in high-dimensional spaces, using spheres, hyperplanes, and wedges, with new topological methods involving Borsuk-Ulam theorems.

Contribution

It introduces novel mass assignment results for spheres, hyperplanes, and wedges, employing new Borsuk-Ulam type theorems on spheres and Stiefel manifolds.

Findings

New mass assignment theorems for spheres, hyperplanes, and wedges

Use of advanced topological tools in mass partition problems

Generalization of classic results to continuous mass distributions

Abstract

In this paper, we generalize classic mass partition results dealing with partitions using spheres, parallel hyperplanes, or axis-parallel wedges to the setting of mass assignments. In a mass assignment problem, we assign mass distributions continuously to all $k$-dimensional subspaces of $\mathbb{R}^d$, and seek to guarantee the existence of a particular subspace in which more masses can be bisected than those by analyzing the problem in $\mathbb{R}^k$. We prove new mass assignment results for spheres, parallel hyperplanes, and axis-parallel wedges. The proof techniques rely on new Borsuk--Ulam type theorems on spheres and Stiefel manifolds.