Lower bounds for weak epsilon-nets and stair-convexity

Understand

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.