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Helly-Type Theorems for Line Transversals to Disjoint Unit Balls

Otfried Cheong, Xavier Goaoc, Andreas Holmsen, Sylvain Petitjean
2007 Discrete & Computational Geometry  
We prove Helly-type theorems for line transversals to disjoint unit balls in R d .  ...  In particular, we show that a family of n ≥ 2d disjoint unit balls in R d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with  ...  Batog for helpful discussions on inflatability.  ... 
doi:10.1007/s00454-007-9022-1 fatcat:c5ydsooky5avdm5rbgrokqfkyy

A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls

Andreas Holmsen, Meir Katchalski, Ted Lewis
2003 Discrete & Computational Geometry  
The proof also uses the recent result on geometric permutations for disjoint unit balls  ...  Let F be a family of disjoint unit balls in R 3 .  ...  . , B n+1 }, with n ≥ n 0 being a family of n + 1 disjoint unit balls in R 3 that has T (n). We shall show that F admits a line transversal, which will imply Theorem 1.  ... 
doi:10.1007/s00454-002-0793-0 fatcat:4slhx5yofvblhf5slqeubyt4gy

Helly-Type Theorems for Line Transversals to Disjoint Unit Balls [chapter]

Otfried Cheong, Xavier Goaoc, Andreas Holmsen, Sylvain Petitjean
Twentieth Anniversary Volume:  
We prove Helly-type theorems for line transversals to disjoint unit balls in R d .  ...  In particular, we show that a family of n 2d disjoint unit balls in R d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with  ...  Batog for helpful discussions on inflatability.  ... 
doi:10.1007/978-0-387-87363-3_12 fatcat:pchtke37are3vc53wvq3b2ofxy

Some Discrete Properties of the Space of Line Transversals to Disjoint Balls [chapter]

Xavier Goaoc
2009 IMA Volumes in Mathematics and its Applications  
We review recent progress in the special case of disjoint Euclidean balls in R d , more precisely the inter-related notions of cone of directions, geometric permutations and Helly-type theorems, and discuss  ...  Attempts to generalize Helly's theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in R d to the structure of the space of lines intersecting  ...  The author is grateful to Boris Aronov, Ciprian Borcea, Otfried Cheong, Olivier Devillers, Hazel Everett, Andreas Holmsen and Sylvain Petitjean for helpful discussions.  ... 
doi:10.1007/978-1-4419-0999-2_3 fatcat:snteyqtjojh6rjbeui6r4ih65m

Hadwiger and Helly-type theorems for disjoint unit spheres [article]

Otfried Cheong, Xavier Goaoc, Andreas Holmsen, Sylvain Petitjean
2007 arXiv   pre-print
We prove Helly-type theorems for line transversals to disjoint unit balls in ^d.  ...  In particular, we show that a family of n ≥ 2d disjoint unit balls in ^d has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with  ...  Batog for helpful discussions on inflatability.  ... 
arXiv:cs/0702039v1 fatcat:fut5f2ced5c3dnohtiuxvag4ya

A Helly-Type Theorem for Hyperplane Transversals to Well-Separated Convex Sets

B. Aronov, J. E. Goodman, R. Pollack, R. Wenger
2001 Discrete & Computational Geometry  
This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.  ...  We say that the collection S is ε-separated if, for every 0 < k < d, any k of the sets can be separated from any other d − k of the sets by a hyperplane more than ε D(S)/2 away from all d of the sets.  ...  For d = 2, our result yields a Helly-type theorem for line transversals to ε-separated collections of pairwise disjoint convex sets in R 2 .  ... 
doi:10.1007/s00454-001-0016-0 fatcat:g2kfdrh5zreufmjepqha6zxbgu

The Harmony of Spheres

Otfried Cheong
2007 Canadian Conference on Computational Geometry  
(For 4 ≥ n ≥ 8, it is still open whether the correct answer is two or three.) The first Helly-type theorem for line transversals to spheres was proven by Holmsen et al. [4] .  ...  In this talk we discuss recent results on transversals to disjoint spheres (balls) in three and more dimensions. Katchalski et al.  ... 
dblp:conf/cccg/Cheong07 fatcat:7limp2lwhjffbettbtgg7jigya

Lower Bounds for Pinning Lines by Balls (Extended Abstract)

Otfried Cheong, Xavier Goaoc, Andreas Holmsen
2009 Electronic Notes in Discrete Mathematics  
We show that the constant 2d is best possible, in any dimension, and derive lower bounds on Helly numbers for sets of line transversals to disjoint balls in arbitrary dimension.  ...  It is known that if n ≥ 2d pairwise disjoint balls in R d have a unique line intersecting them in a given order ≺, one can always remove a ball so that remains the only line intersecting the balls in the  ...  The corresponding Helly number, for transversals to disjoint unit balls in R d , is known to be between 5 and 4d − 1.  ... 
doi:10.1016/j.endm.2009.07.094 fatcat:phwmwb5j2rf6rpm4jnh4oejfhi

Discrete and Lexicographic Helly-Type Theorems

Nir Halman
2007 Discrete & Computational Geometry  
We define three new types of Helly theorems: discrete Helly theorems-where the common point should belong to an a-priori given set, lexicographic Helly theorems-where the common point should not be lexicographically  ...  We study the relations between the different types of the Helly theorems. We obtain several new discrete and lexicographic Helly numbers.  ...  Acknowledgements The author would like to thank Branko Grünbaum for inspiring discussions and an anonymous referee for many useful comments which greatly improved the presentation of this paper.  ... 
doi:10.1007/s00454-007-9028-8 fatcat:xfxzbfvfbzdbbeactazvek5jda

The Katchalski-Lewis Transversal Problem in Rn

Andreas Holmsen
2007 Discrete & Computational Geometry  
Also we give two constructions of families of pairwise disjoint translates of the unit ball in R 3 which answer some related questions. * Supported by the Research Council of Norway. 342 A. Holmsen  ...  line.  ...  Acknowledgments Special thanks to Boris Aronov for many helpful discussions. Also the construction of Proposition 7 was found independently by B. Aronov, V. Koltun, and M. Sharir.  ... 
doi:10.1007/s00454-006-1291-6 fatcat:bmpbwhmlabe4vfetglrskmmrgm

Line Transversals to Unit Disks

Kaiser
2002 Discrete & Computational Geometry  
We show that if every three members of a finite disjoint family of unit disks in the plane have a line transversal, then there is a line transversal to all except at most 12 disks in the family.  ...  We derive an analogous result for translates of a general compact convex set, with the constant equal to 47.  ...  Acknowledgments I am indebted to Boris Aronov, Wlodek Kuperberg and two anonymous referees whose comments were helpful in improving both the results and the presentation of this paper.  ... 
doi:10.1007/s00454-002-2821-5 fatcat:efqfo42ykbbyxgaaqxggdy374y

No Helly Theorem for Stabbing Translates by Lines in R 3

Andreas Holmsen, Jiri Matousek
2004 Discrete & Computational Geometry  
For each n > 2 we construct a convex body K ⊂ R 3 and a finite family F of disjoint translates of K such that any n − 1 members F admit a line transversal, but F has no line transversal.  ...  [6] shows that there is a finite Helly number 406 A. Holmsen and J. Matoušek for line transversals to families of disjoint unit balls in R 3 .  ...  There is no Hadwiger-type theorem for line transversals to families of disjoint translates in R 3 . 3.  ... 
doi:10.1007/s00454-003-0796-5 fatcat:vnizeoljtnaypdyr46vfkiltyu

A Helly-type theorem for higher-dimensional transversals

Boris Aronov, Jacob E. Goodman, Richard Pollack
2002 Computational geometry  
It is the first Helly-type theorem known for transversals of dimension between 1 and d − 1.  ...  This result generalizes a theorem of Hadwiger(-Danzer-Grünbaum-Klee) on line transversals for an unbounded family of compact convex sets.  ...  of Helly type were found for both line transversals and hyperplane transversals to collections restricted by size or shape, such as unit balls, parallelepipeds, translates of a single convex body, etc  ... 
doi:10.1016/s0925-7721(01)00025-6 fatcat:odkjicyzercupizvhyk4fgz2hu

Page 2227 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
disjoint unit balls.  ...  Carla Peri (l-SACM-GM,; Milan) 2004¢:52010 52A35 Holmsen, Andreas (N-BERG; Bergen) ; Katchalski, Meir (IL-TECHM; Haifa); Lewis, Ted (3-AB-MS; Edmonton, AB) A Helly-type theorem for line transversals to  ... 

Helly-Type Theorems and Geometric Transversals [chapter]

Rephael Wenger
2004 Handbook of Discrete and Computational Geometry, Second Edition  
HADWIGER'S TRANSVERSAL THEOREM In 1935, Vincensini asked if there is a Helly-type theorem for line transversals to a family A of convex sets in R 2 .  ...  CONVEXITY ON THE AFFINE GRASSMANNIAN There is no Helly-type theorem for convex sets of kats in R d since such a theorem would be equivalent to a Helly-type theorem for k-transversals in R d .  ... 
doi:10.1201/9781420035315.ch4 fatcat:x3epiiq2kbf6nenkswsfaqmsdi
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