Ham-Sandwich Cuts and Center Transversals in Subspaces

Abstract

The Ham-Sandwich theorem is a well-known result in geometry. It states that any d mass distributions in ${\mathbb {R}}^d$ can be simultaneously bisected by a hyperplane. The result is tight, that is, there are examples of $d+1$ mass distributions that cannot be simultaneously bisected by a single hyperplane. In this paper we will study the following question: given a continuous assignment of mass distributions to certain subsets of ${\mathbb {R}}^d$, is there a subset on which we can bisect more masses than what is guaranteed by the Ham-Sandwich theorem? We investigate two types of subsets. The first type are linear subspaces of ${\mathbb {R}}^d$, i.e., k-dimensional flats containing the origin. We show that for any continuous assignment of d mass distributions to the k-dimensional linear subspaces of ${\mathbb {R}}^d$, there is always a subspace on which we can simultaneously bisect the images of all d assignments. We extend this result to center transversals, a generalization of Ham-Sandwich cuts. As for Ham-Sandwich cuts, we further show that for $d-k+2$ masses, we can choose $k-1$ of the vectors defining the k-dimensional subspace in which the solution lies. The second type of subsets we consider are subsets that are determined by families of n hyperplanes in ${\mathbb {R}}^d$. Also in this case, we find a Ham-Sandwich-type result. In an attempt to solve a conjecture by Langerman about bisections with several cuts, we show that our underlying topological result can be used to prove this conjecture in a relaxed setting.