  A set N is called a "weak epsilon-net" (with respect to convex sets) for a
finite set X in R^d if N intersects every convex set that contains at least
epsilon*|X| points of X. For every fixed d>=2 and every r>=1 we construct sets
X in R^d for which every weak (1/r)-net has at least Omega(r log^{d-1} r)
points; this is the first superlinear lower bound for weak epsilon-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 epsilon-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 upper bounds by Alon, Kaplan, Nivasch, Sharir, and
Smorodinsky (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(t^2 / (n^3 log (n^3/t))) triangles of
T.
