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Point Selections and Weak ε-Nets for Convex Hulls

Published online by Cambridge University Press:  12 September 2008

Noga Alon ,
Imre Bárány ,
Zoltán Füredi  and
Daniel J. Kleitman
Noga Alon
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel, and Bellcore, Morristown, NJ 07960, USA. e-mail: [email protected]
Imre Bárány
Affiliation:
Cowles Foundation, Yale University, New Haven, CT 06520, USA, and Courant Institute, New York University, New York, NY 10009, USA. e-mail: [email protected]
Zoltán Füredi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences POB 127, 1364 Budapest, Hungary, and Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. e-mail: [email protected]
Daniel J. Kleitman
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA. e-mail: [email protected]
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Abstract

One of our results: let X be a finite set on the plane, 0 < ε < 1, then there exists a set F (a weak ε-net) of size at most 7/ε2 such that every convex set containing at least ε|X| elements of X intersects F. Note that the size of F is independent of the size of X.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 3 , September 1992 , pp. 189 - 200
Copyright
Copyright © Cambridge University Press 1992

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References

[1]Alon, N. and Kleitman, D. J. (to appear) Piercing convex sets and the Hadwiger-Debrunner (p, q)-problem. Adv. Math.Google Scholar
[2]Aronov, B., Chazelle, B., Edelsbrunner, H., Guibas, L. J., Sharir, M. and Wenger, R. (1991) Points and triangles in the plane and halving planes in space. Discrete Comp. Geom. 6 435–442.CrossRefGoogle Scholar
[3]Bárány, I. (1982) A generalization of Caratheodory's theorem. Discrete Math. 40 141–152.Google Scholar
[4]Bárány, I., Füredi, Z. and Lovász, L. (1990) On the number of halving planes. Combinatorica 10 175–183.Google Scholar
[5]Bárány, I. and Larman, G. D. (to appear) A coloured version of Tverberg's theorem. J. London Math. Soc.Google Scholar
[6]Beck, J. and Chen, W. (1987) Irregularities of distributions. Cambridge Tracts in Math. 89, Cambridge University Press.Google Scholar
[7]Boros, E. and Füredi, Z. (1984) The number of triangles covering the center of an n-set. Geom Dedicata 17 69–77.CrossRefGoogle Scholar
[8]Capoyleas, V. (1991) An upper bound for weak ε-nets of points in convex position: preliminary version,(manuscript).Google Scholar
[9]Chernoff, H. (1952) A measure of asymptotic efficiency for tests of an hypothesis based on the sum of observations. Ann. Math. Statist. 23 497–507.CrossRefGoogle Scholar
[10]Erdős, P. (1964) On extremal problems of graphs and generalized graphs. Israel J. Math. 2 183–190.CrossRefGoogle Scholar
[11]Erdős, P. and Simonovits, M. (1983) Supersaturated graphs and hypergraphs. Combinatorica 3 181–192.CrossRefGoogle Scholar
[12]Jaromczyk, J. W. and Światek, G. (1990) The optimal constant for the colored version of Tverberg's theorem. DIMACS Technical Report 90–6.Google Scholar
[13]Katchalski, M. and Meir, A. (1988) On empty triangles determined by points in the plane. Acta Math Hungar. 51 323–28.Google Scholar
[14]Lováasz, L. (1971) On the number of halving lines. Annales Univ Sci Budapest, Eötvös 14 107–108.Google Scholar
[15]Megiddo, N. (1985) Partitioning with two lines in the plane. J. Algorithms 6 430–433.Google Scholar
[16]Tverberg, H. (1966) A generalization of Radon's theorem. J. London Math. Soc. 41 123–128.Google Scholar
[17]Welzl, E. (1990) Lecture at a DIMACS conference,January 1990.Google Scholar
[18]Yao, F. F. (1983) A 3-space partition and its applications. Proc. 15th Annual ACM STOC, ACM Press, New York, 258–263.Google Scholar
[19]ZŽivaljević, R. T. and Vrećica, S. T. (to appear) The coloured Tverberg problem and complexes of injective functions. J. Comb. Theory, AGoogle Scholar