Discrete & Computational Geometry

, Volume 7, Issue 2, pp 163–173

Almost tight bounds forɛ-Nets


  • János Komlós
    • Department of MathematicsRutgers University
  • János Pach
    • Mathematical InstituteHungarian Academy of Sciences
    • Courant Institute of Mathematical SciencesNew York University
  • Gerhard Woeginger
    • Institut für MathematikTechnische Universität Graz

DOI: 10.1007/BF02187833

Cite this article as:
Komlós, J., Pach, J. & Woeginger, G. Discrete Comput Geom (1992) 7: 163. doi:10.1007/BF02187833


Given any natural numberd, 0<ɛ<1, letfd(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.

Copyright information

© Springer-Verlag New York Inc. 1992