On the Helly Number for Hyperplane Transversals to Unit Balls
نویسندگان
چکیده
منابع مشابه
On the Helly Number for Hyperplane Transversals to Unit Balls
We prove some results about the Hadwiger problem of nding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d-dimensional Euclidean space. These include (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks|thus correcting...
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We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...
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INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 ≤ k < d, a k-dimensional affine subspace that intersects every member of F is called a ktransversal to F . Typically, we are interested in necessary and sufficient conditions that guarantee the existence of a k-transversal...
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We generalize the Hadwiger(-Danzer-Grünbaum-Klee) theorem on line transversals for an unbounded family of compact convex sets to the case of transversal planes of arbitrary dimension. This is the first Helly-type theorem known for transversals of dimension between 1 and d− 1.
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Let D be a set of n pairwise disjoint unit balls in R and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at least m points from P . This generalizes the notion of halving hyperplanes, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for ins...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2000
ISSN: 0179-5376
DOI: 10.1007/s004540010024