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 $$S_{k-1} \subset S_k \subset \ldots \subset S_{d-1} \subset {\mathbb {R}}^d$$ 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.