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Discrete & Computational Geometry

, Volume 13, Issue 1, pp 1–15 | Cite as

Improved bounds on weak ε-nets for convex sets

  • B. Chazelle
  • H. Edelsbrunner
  • M. Grigni
  • L. Guibas
  • M. Sharir
  • E. Welzl
Article

Abstract

LetS be a set ofn points in ℝ d . A setW is aweak ε-net for (convex ranges of)S if, for anyTS containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size\(O((1/\varepsilon ^d )\log ^{\beta _d } (1/\varepsilon ))\), whereβ2=0,β3=1, andβ d ≈0.149·2d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas.

Keywords

Interval Tree Hyperbolic Plane Hyperbolic Geometry Hyperbolic Distance Improve Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • B. Chazelle
    • 1
  • H. Edelsbrunner
    • 2
  • M. Grigni
    • 1
  • L. Guibas
    • 3
    • 4
  • M. Sharir
    • 5
    • 6
  • E. Welzl
    • 7
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA
  2. 2.Computer Science DepartmentUniversity of IllinoisUrbanaUSA
  3. 3.DEC Systems Research Center, Laboratory for Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA
  4. 4.Computer Science DepartmentStanford UniversityStanfordUSA
  5. 5.School of Mathematical SciencesTel Aviv UniversityRamet AvivIsrael
  6. 6.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  7. 7.Institut für InformatikFreie Universität BerlinBerlinGermany

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