A Constant Bound for Geometric Permutations of Disjoint Unit Balls
نویسندگان
چکیده
منابع مشابه
Geometric permutations of disjoint unit spheres
We show that a set of n disjoint unit spheres in R admits at most two distinct geometric permutations if n ≥ 9, and at most three if 3 ≤ n ≤ 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R: if any subset of size 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.
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(i) We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in IR d , is (n d?1). This improves substantially the general upper bound of O(n 2d?2) given in 11]. (ii) We show that the maximum number of geometric permutations of a suuciently large collection of pairwise disjoint unit discs in the plane is 2, improving a...
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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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ژورنال
عنوان ژورنال: Discrete and Computational Geometry
سال: 2003
ISSN: 0179-5376
DOI: 10.1007/s00454-002-2828-y