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 ℝ𝑑 . Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces 𝑆𝑘−1⊂𝑆𝑘⊂…⊂𝑆𝑑−1⊂ℝ𝑑 such that 𝑆𝑖 bisects all the mass assignments on 𝑆𝑖+1 for every i. We show it is possible to do so whenever the number of mass assignments of dimensions (𝑘,…,𝑑) is a permutation of (𝑘,…,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 ℝ𝑑 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.