Lower bounds for weak epsilon-nets and stair-convexity

A set N ⊂ ℝ^d is called a weak ε-net (with respect to convex sets) for a finite X ⊂ ℝ^d if N intersects every convex set C with |X ∩ C| ≥ ε|X|. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ ℝ^d for which every weak 1/r-net has at least Ω (r log^d-1 r) points; this is the first superlinear lower bound for weak ε-nets in a fixed dimension. The construction is a stretched grid, i.e., the Cartesian product of d suitable fast-growing finite sequences, and convexity in this grid can be analyzed using stair-convexity, a new variant of the usual notion of convexity. We also consider weak ε-nets for the diagonal of our stretched grid in R ^d, d ≥ 3, which is an "intrinsically 1- dimensional" point set. In this case we exhibit slightly superlinear lower bounds (involving the inverse Ackermann function), showing that the upper bounds by Alon et al. (2008) are not far from the truth in the worst case. Using the stretched grid we also improve the known upper bound for the so-called second selection lemma in the plane by a logarithmic factor: We obtain a set T of t triangles with vertices in an n-point set in the plane such that no point is contained in more than O ( t²/(n³ log n³ t )) triangles of T.