Optimal lower bounds for epsilon-nets for lines in the plane

Abstract

We prove that, for arbitrarily small positive $\varepsilon$, there is a finite planar point set $P$ such that every $\varepsilon$-net for the range space induced on $P$ by straight lines has cardinality $Ω\bigl(1/\varepsilon \cdot \log(1/\varepsilon)\bigr)$. This matches the classical upper bound for range spaces with bounded VC-dimension due to Haussler and Welzl and confirms a prediction of Alon.