Some new bounds for Epsilon-nets

Authors:
János Pach;Gerhard Woeginger
Affiliations:
Mathematical Institute of the Hungarian, Academy of Sciences, Budapest, H-1364, Pf. 127, Courant Institute of Mathematical Sciences, New York University, New York;Institut für Mathematik, Technische Universität Graz, Kopernikusgasse 24, A-8010 Graz, Austria
Venue:
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Year:
1990

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Abstract

Given any natural number d, 0 &egr; d(&egr;) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an &egr;-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒd(&egr;) 1/48 d/&egr; log 1/ &egr; which is not far from being optimal, if d is fixed and &egr; → 0. Further, we prove that ƒ1(&egr;) = max(2,⌈1/&egr;⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.