Orthogonal ham-sandwich theorem in R3

Authors:
Sergey Bereg
Affiliations:
University of Texas at Dallas, Richardson, TX
Venue:
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Year:
2010

Citing 7
Cited 0

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Abstract

The ham-sandwich theorem states that, given d ≥ 2 measures in Rd, it is possible to divide all of them in half with a single (d - 1)-dimensional hyperplane. We study an orthogonal version of the ham-sandwich theorem and define an orthogonal cut using at most d hyperplanes orthogonal to coordinate axes. For example, a hyperplane orthogonal to a coordinate axis and the boundary of an orthant are orthogonal cuts. We prove that any three measures in R3 can be divided in half each with a single orthogonal cut. Applied to point measures, it implies that any three finite sets of points in R3 can be simultaneously bisected by an orthogonal cut. We present an algorithm for computing an orthogonal ham-sandwich cut in O(n log n) time.