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Ham-Sandwich Cuts and Center Transversals in Subspaces

Author Patrick Schnider

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LIPIcs.SoCG.2019.56.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Ham-Sandwich cuts
  • center transversal
  • topological methods

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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 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 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 subsets we 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.

Cite As Get BibTex

Patrick Schnider. Ham-Sandwich Cuts and Center Transversals in Subspaces. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.56

Author Details

Patrick Schnider
  • Department of Computer Science, ETH Zürich

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