Bisections of mass assignments using flags of affine spaces

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 $S_{k-1}\subset S_k\subset \dots \subset S_{d-1}\subset {\mathbb{R}}^d$ 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 $(k,\dots ,d)$ is a permutation of $(k,\dots ,d)$. 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.

17 pages, 7 figures