Cutting dense point sets in half Journal Article

Author(s): Edelsbrunner, Herbert; Valtr, Pavel; Welzl, Emo
Article Title: Cutting dense point sets in half
Abstract: A halving hyperplane of a set S of n points in R(d) contains d affinely independent points of S so that equally many of the points off the hyperplane lie in each of the two half-spaces. We prove bounds on the number of halving hyperplanes under the condition that the ratio of largest over smallest distance between any two points is at most delta n(1/d), delta some constant. Such a set S is called dense. In d = 2 dimensions the number of halving lines for a dense set can be as much as Omega(n log n), and it cannot exceed O (n(5/4)/log* n). The upper bound improves over the current best bound of O (n(3/2)/log* n) which holds more generally without any density assumption. In d = 3 dimensions we show that O (n(7/3)) is an upper bound on the number of halving planes for a dense set, The proof is based on a metric argument that can be extended to d greater than or equal to 4 dimensions, where it leads to O (n(d-2/d)) as an upper bound for the number of halving hyperplanes.
Journal Title: Discrete & Computational Geometry
Volume: 17
Issue 3
ISSN: 0179-5376
Publisher: Springer  
Date Published: 1997-04-01
Start Page: 243
End Page: 255
Sponsor: Partially supported by the National Science Foundation, under Grant ASC-9200301 and the Alan T. Waterman award, Grant CCR-9118874.
DOI: 10.1007/PL00009291
Open access: no
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