Bredon cohomology methods in mass partition problems on spheres

TL;DR

This paper introduces a novel application of RO(G)-graded Bredon cohomology to mass partition problems on spheres, providing a new topological framework that extends classical methods and rederives recent results.

Contribution

It applies Bredon cohomology to mass assignment problems, offering a flexible topological approach that broadens the scope of classical mass partition techniques.

Findings

Reproved a recent mass partition result using Bredon cohomology.

Demonstrated the potential of the framework for broader applications.

Extended classical methods with a new topological perspective.

Abstract

We apply $\mathrm{RO}(G)$-graded Bredon cohomology to mass assignment problems, extending classical mass partition methods. Within this framework, we reprove a recent result of Lessure and Sober\'on: for $n+1$ mass assignments on $k$-dimensional affine subspaces of $\mathbb{R}^n$, there exists a $k$-subspace containing a sphere that simultaneously bisects all measures. This approach highlights a flexible topological framework with potential for broader applications.