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.
Highlights
We show that for any continuous assignment of d mass distributions to the k-dimensional linear subspaces of Rd, 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
Summary
The famous Ham-Sandwich theorem (see e.g. [15, 19], Chapter 21 in [20]) is a central result in geometry that initiated a significant amount of research on several ways to partition mass distributions. The Ham-Sandwich theorem says that on every subspace we can simultaneously bisect the images of k mass assignments. For k = n we get that there is always a subspace on which we can simultaneously bisect d images of mass assignments The second generalization of the Ham-Sandwich theorem that we investigate in this paper considers bisections with several cuts, where the masses are distributed into two parts according to a natural 2-coloring of the induced arrangement. Let Gd−1(Rd)n be the space of all sets of n hyperplanes containing the origin (i.e., linear subspaces) in Rd. Similar to before, we define a mass assignment μ on Gd−1(Rd)n as a continuous assignment Gd−1(Rd)n → Md, where Md again denotes the space of all d-dimensional mass distributions. The standard reference for this concept is the classic book by Milnor and Stasheff [16]
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