Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (1/epsilon) (Pach and Tardos).

The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.