According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size. 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 ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size, 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, in which the size of the smallest ε-nets is. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.
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