Ham-Sandwich Cuts and Center Transversals in Subspaces
METADATA ONLY
Loading...
Author / Producer
Date
2020-12
Publication Type
Journal Article
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
Abstract
The Ham-Sandwich theorem is a well-known result in geometry. It states that any d mass distributions in 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 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 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 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 subsetswe consider are subsets that are determined by families of n hyperplanes in 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.
Permanent link
Publication status
published
External links
Editor
Book title
Journal / series
Volume
64 (4)
Pages / Article No.
1192 - 1209
Publisher
Springer
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Ham-Sandwich theorem; Center transversal theorem; Topological methods; Mass partitions
Organisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)