tight lower bounds for the size of epsilon-nets
According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ?-net of size O(1/? log 1/?). Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ?-nets is superlinear in 1/?, were found by Alon (2010). In his examples, every ?-net is of size ?(1/?g(1/?)), where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VC-dimension 2, in which the size of the smallest ?-nets is ?(1/? log 1/?). We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ?-nets is ?(1/? log log 1/?). By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.
epsilon-net geometry