Abstract
The HamSandwich theorem is a wellknown 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 abstract 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 HamSandwich theorem?
We investigate two types of subsets. The first type are linear subspaces of R^d, i.e., kdimensional flats containing the origin. We show that for any continuous assignment of d mass distributions to the kdimensional 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 HamSandwich cuts. As for HamSandwich cuts, we further show that for dk+2 masses, we can choose k1 of the vectors defining the kdimensional subspace in which the solution lies.
The second type of subsets we consider are subsets that are determined by families of n hyperplanes in R^d. Also in this case, we find a HamSandwichtype 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.
BibTeX  Entry
@InProceedings{schnider:LIPIcs:2019:10460,
author = {Patrick Schnider},
title = {{HamSandwich Cuts and Center Transversals in Subspaces}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {56:156:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771047},
ISSN = {18688969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10460},
URN = {urn:nbn:de:0030drops104609},
doi = {10.4230/LIPIcs.SoCG.2019.56},
annote = {Keywords: HamSandwich cuts, center transversal, topological methods}
}
Keywords: 

HamSandwich cuts, center transversal, topological methods 
Collection: 

35th International Symposium on Computational Geometry (SoCG 2019) 
Issue Date: 

2019 
Date of publication: 

11.06.2019 