Lower bounds for weak epsilon-nets and stair-convexity

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.

💡 Research Summary

The paper addresses the long‑standing open problem of establishing super‑linear lower bounds for weak ε‑nets in fixed dimension. A weak ε‑net for a finite point set X⊂ℝ^d is a set N that meets every convex set containing at least ε·|X| points of X. While linear lower bounds Ω(r) (with ε=1/r) are known for any dimension, super‑linear bounds were previously obtained only when the dimension grows with r. The authors break this barrier by constructing, for every fixed d≥2 and every integer r≥1, a point set X in ℝ^d for which any weak (1/r)‑net must contain at least Ω(r·log^{d‑1} r) points. This is the first super‑linear lower bound that holds in a constant dimension.

Stretched grid construction.
The core of the construction is a “stretched grid”, i.e. the Cartesian product of d rapidly increasing integer sequences A_1,…,A_d. Each sequence grows so fast that the ratios between consecutive elements are huge, which forces any large subset of the grid to be spread out across many “layers” when projected onto each coordinate axis. The authors partition the grid into r layers; each layer contains Θ(log^{d‑1} r) points, and the layers are mutually disjoint in every coordinate projection.

Stair‑convexity.
To analyze convexity inside this highly anisotropic grid the authors introduce a new geometric notion called stair‑convexity. A set S⊂ℝ^d is stair‑convex if for any two points p,q∈S the monotone (non‑decreasing in each coordinate) “stair” path from p to q lies entirely in S. Equivalently, S can be described as the union of axis‑parallel boxes that respect the coordinate order. This notion captures the combinatorial structure of the stretched grid much more faithfully than ordinary Euclidean convexity: a large stair‑convex set must contain a full interval in each coordinate, which translates into a strong combinatorial constraint on the distribution of points across layers.

Deriving the lower bound.
Given ε=1/r, any weak (1/r)‑net N must intersect every stair‑convex set that contains at least one full layer (because such a set automatically contains ≥ε·|X| points). However, a single point of N can intersect stair‑convex sets from only O(1) different layers, because covering two layers would require the point’s coordinates to simultaneously belong to two disjoint intervals on at least one axis, which is impossible. Consequently, to hit all r layers N must contain at least Ω(r·log^{d‑1} r) points. This argument yields the desired lower bound and matches the known upper bound O(r·log^{d‑1} r) up to constant factors, thereby settling the asymptotic order for fixed d.

Diagonal of the stretched grid (intrinsically 1‑dimensional case).
The authors also study the set D consisting of the diagonal points {(a_1(k),…,a_d(k)) | k=1,…,m}. Although D is essentially one‑dimensional, the surrounding stretched grid forces weak ε‑nets for D to behave similarly to the full d‑dimensional case. By a refined counting argument they prove a lower bound of Ω(r·α(r)), where α(·) is the inverse Ackermann function. This matches, up to constants, the upper bound O(r·α(r)) proved by Alon, Kaplan, Nivasch, Sharir, and Smorodinsky (2008). Hence even for “thin” point sets the worst‑case size of weak ε‑nets can be near‑optimal only up to the extremely slowly growing α(·).

Application to the second selection lemma.
Finally, the stretched grid technique is applied to improve the planar second selection lemma. The lemma concerns a set P of n points in the plane and a family T of t triangles with vertices in P; it asks for the smallest possible maximum number of triangles from T that can contain a single point of the plane. The classic bound is O(t²/n³). By arranging the points of P on a 2‑dimensional stretched grid and selecting triangles that respect the stair‑convex structure, the authors construct a family T for which every point lies in at most O(t²/(n³·log(n³/t))) triangles. This improves the previous bound by a logarithmic factor, showing that the combinatorial complexity of triangle containment can be reduced using the same geometric insight that yielded the ε‑net lower bounds.

Impact and outlook.
The paper introduces two powerful tools—stretched grids and stair‑convexity—that together enable a precise combinatorial analysis of convexity in highly non‑uniform point configurations. The super‑linear lower bound for weak ε‑nets resolves a major open question in discrete geometry and demonstrates that the known upper bounds are essentially tight in fixed dimension. The diagonal result shows that even “low‑dimensional” subsets can inherit the full difficulty of the high‑dimensional problem. Moreover, the improvement of the second selection lemma illustrates that these ideas have broader applicability beyond ε‑nets, potentially influencing range‑search data structures, geometric discrepancy theory, and the design of efficient approximation algorithms for geometric hitting set problems. Future work may explore further variants of stair‑convexity, extend the stretched‑grid methodology to other geometric range families, or seek tight bounds for related concepts such as weak ε‑approximations and ε‑samples.