Discrete & Computational Geometry

, Volume 7, Issue 2, pp 163–173 | Cite as

Almost tight bounds forɛ-Nets

  • János Komlós
  • János Pach
  • Gerhard Woeginger


Given any natural numberd, 0<ɛ<1, letf d (ɛ) denote the smallest integerf such that every range space of Vapnik-Chervonenkis dimensiond has anɛ-net of size at mostf. We solve a problem of Haussler and Welzl by showing that ifd≥2, then
$$d - 2 + \frac{2}{{d + 2}} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{f_d (\varepsilon )}}{{(1/\varepsilon )\log (1/\varepsilon )}} \leqslant d.$$
Further, we prove thatf1(ɛ)=max(2, ⌌ 1/ɛ ⌍−1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.


Maximal Element Discrete Comput Geom Computational Geometry Range Space Simple Polygon 
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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • János Komlós
    • 1
  • János Pach
    • 2
    • 3
  • Gerhard Woeginger
    • 4
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Mathematical InstituteHungarian Academy of SciencesBudapestHungary
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.Institut für MathematikTechnische Universität GrazGrazAustria

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