Hostname: page-component-6766d58669-76mfw Total loading time: 0 Render date: 2026-05-20T04:15:19.991Z Has data issue: false hasContentIssue false
  • English
  • Français

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: noga@math.tau.ac.il
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: h2923bar@ella.hu
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: zoltan@math.uiuc.edu
Daniel J. Kleitman
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA. e-mail: djk@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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