Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
نویسندگان
چکیده
منابع مشابه
Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
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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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...
متن کاملHadwiger and Helly-Type Theorems for Disjoint Unit Spheres in R
Let S be an ordered set of disjoint unit spheres in R. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S.
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2007
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-007-9022-1