Epsilon-nets for halfplanes

Abstract

Given some finite point set P in the plane and some real ε, 0<ε<1, we want to colour a minimal subset of P red, such that the following holds: every open halfplane that contains more than ε · ¦P¦ of the points in P also contains at least one red point.

It is shown that it always suffices to colour [2/ε]-1 points red (independent of the size of P). If ε<2/3, we can choose these [2/ε]-1 points among the extreme points of P. If all red points must be extreme, our solution is optimal and it can be found in O(nlogn) time. If the red points are allowed to be any elements of P, our result is almost optimal: There are point sets requiring at least 2[1/ε]-2 red points. The both bounds differ at most by one.