An Improved Bound for Weak Epsilon-nets in the Plane

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We show that for any finite point set P in the plane and epsilon > 0 there exist O(1/epsilon(3/2+gamma)) points in R-2, for arbitrary small gamma > 0, that pierce every convex set K with vertical bar K boolean AND P vertical bar = epsilon vertical bar P vertical bar. This is the first improvement of the bound of O(1/epsilon 2) that was obtained in 1992 by Alon, Barany, Furedi, and Kleitman for general point sets in the plane.
Original languageEnglish
Article number32
Pages (from-to)1-35
JournalJournal of the ACM
Issue number5
StatePublished - 1 Oct 2022


  • Epsilon-nets
  • VC-dimension
  • convex sets
  • piercing numbers
  • geometric transversals
  • arrangements of lines

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