Lower Bounds to Helly Numbers of Line Transversals to Disjoint Congruent Balls

Abstract : A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957.
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Submitted on : Thursday, September 16, 2010 - 12:21:32 PM
Last modification on : Monday, June 24, 2019 - 12:32:04 PM

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Otfried Cheong, Xavier Goaoc, Andreas Holmsen. Lower Bounds to Helly Numbers of Line Transversals to Disjoint Congruent Balls. Israël Journal of Mathematics, The Hebrew University Magnes Press, 2012, 190 (1), pp.213-228. ⟨https://link.springer.com/article/10.1007%2Fs11856-011-0179-1⟩. ⟨inria-00518035⟩

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