Bisections of mass assignments using flags of affine spaces

TL;DR

This paper generalizes the ham sandwich theorem to mass assignments on affine subspaces using advanced topological methods, establishing new bisecting theorems for complex geometric configurations.

Contribution

It introduces a novel extension of the ham sandwich theorem for mass assignments across affine subspaces, utilizing recent topological results on Stiefel manifolds.

Findings

Established conditions for bisecting multiple mass assignments of different dimensions.

Extended the central transversal theorem to new geometric contexts.

Provided dynamic ham sandwich theorems for moving point families.

Abstract

We use recent extensions of the Borsuk--Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A $k$-dimensional mass assignment continuously imposes a measure on each $k$-dimensional affine subspace of $\mathbb{R}^d$. Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces $S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset \mathbb{R}^d$ such that $S_i$ bisects all the mass assignments on $S_{i+1}$ for every $i$. We show it is possible to do so whenever the number of mass assignments of dimensions $(k,\ldots,d)$ is a permutation of $(k,\ldots,d)$. We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of $(d-k)$-dimensional affine spaces of $\mathbb{R}^d$ using a $(k-1)$-dimensional affine subspace contained in some translate of a fixed $k$-dimensional affine space. For $k=d-1$, there results can be interpreted as dynamic ham sandwich theorems for families of moving points.