A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls |
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| Authors: | Holmsen Katchalski Lewis |
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| Affiliation: | (1) Department of Mathematics, University of Bergen, Johannes Brunsgt. 12, 5008 Bergen, Norway andreash@mi.uib.no , NO;(2) Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel meirk@techunix.technion.ac.il , IL;(3) Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, Alberta, Canada T6G 2G1 tlewis@math.ualberta.ca, CA |
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| Abstract: | Abstract. Let F be a family of disjoint unit balls in R 3 . We prove that there is a Helly-number n 0 ≤ 46 , such that if every n 0 members of F ( | F | ≥ n 0 ) have a line transversal, then F has a line transversal. In order to prove this we prove that if the members of F can be ordered in a way such that every 12 members of F are met by a line consistent with the ordering, then F has a line transversal. The proof also uses the recent result on geometric permutations for disjoint unit balls by Katchalski, Suri, and Zhou. |
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