Small Strong Epsilon Nets

Let P be a set of n points in $\mathbb{R}^d$. A point x is said to be a centerpoint of P if x is contained in every convex object that contains more than $dn\over d+1$ points of P. We call a point x a strong centerpoint for a family of objects $\mathcal{C}$ if $x \in P$ is contained in every object $C \in \mathcal{C}$ that contains more than a constant fraction of points of P. A strong centerpoint does not exist even for halfspaces in $\mathbb{R}^2$. We prove that a strong centerpoint exists for axis-parallel boxes in $\mathbb{R}^d$ and give exact bounds. We then extend this to small strong $\epsilon$-nets in the plane and prove upper and lower bounds for $\epsilon_i^\mathcal{S}$ where $\mathcal{S}$ is the family of axis-parallel rectangles, halfspaces and disks. Here $\epsilon_i^\mathcal{S}$ represents the smallest real number in $[0,1]$ such that there exists an $\epsilon_i^\mathcal{S}$-net of size i with respect to $\mathcal{S}$.

💡 Research Summary

The paper introduces the notion of a strong centerpoint and extends it to the concept of strong ε‑nets for specific geometric range families. A classic centerpoint in ℝ^d is a point that lies inside every convex set containing more than dn/(d+1) points of a given point set P. By contrast, a strong centerpoint must be an actual element of P and must be contained in every range from a prescribed family ℂ that captures a fixed constant fraction of P. This additional restriction makes the existence of strong centerpoints far less guaranteed than that of ordinary centerpoints.

The authors first demonstrate that strong centerpoints do not exist for halfspaces even in the plane, confirming earlier intuition that the requirement is too stringent for many natural range families. They then turn to axis‑parallel boxes (in ℝ^d, axis‑parallel hyper‑rectangles) and prove that a strong centerpoint always exists for this family. The construction is elementary: sort the points along each coordinate axis and pick a point that is median in every dimension. Such a point is guaranteed to belong to any axis‑parallel box that contains more than n/2^d points of P. The authors show that the bound ε₁^ℬ = 1/2^d is tight by presenting a matching lower‑bound construction where 2^d−1 points occupy the corners of a hyper‑cube and one point sits at the centre. Any box covering a fraction larger than 1/2^d can avoid the centre, proving optimality.

Having established the existence of a strong centerpoint for boxes, the paper proceeds to study strong ε‑nets of size i. An ε‑net of size i for a range family ℰ is a subset Q⊆P of cardinality i such that any range C∈ℰ containing at least ε·|P| points of P also contains at least one point of Q. The authors denote by ε_i^ℰ the smallest ε for which such a net of size i exists. Their focus is on three planar families: axis‑parallel rectangles ℛ, halfspaces ℋ, and disks 𝔻.

1. Axis‑parallel rectangles (ℛ).
   Using the strong centerpoint result (ε₁^ℛ = 1/4) as a base, the authors develop a recursive scheme: each additional point roughly halves the uncovered fraction. They prove an upper bound ε_i^ℛ ≤ 1/2^{i+1}. For the lower bound, they construct a grid of 2^{i+1}−1 points arranged so that any rectangle covering more than 1/2^{i+1} of the points must contain a grid point, establishing ε_i^ℛ ≥ 1/2^{i+1}. Consequently, ε_i^ℛ = 1/2^{i+1} is exact for all i≥1, showing that strong ε‑nets for rectangles shrink exponentially with the net size.

2. Halfspaces (ℋ).
   Although a strong centerpoint does not exist for halfspaces, the authors show that ε₁^ℋ = 1/2 is optimal: any halfspace containing more than half the points must intersect a suitably chosen single point. For larger i, they prove ε_i^ℋ = 1/(i+1). The upper bound follows from placing points on a circle and selecting i points spaced evenly; any halfspace covering more than 1/(i+1) of the points inevitably captures one of the selected points. The matching lower bound is obtained by a symmetric circular arrangement that allows a halfspace to avoid the chosen i points while still covering exactly 1/(i+1) of the total. Hence the bound is tight.

3. Disks (𝔻).
   Disks are more challenging because of rotational symmetry. The authors obtain an upper bound ε_i^𝔻 ≤ 2/(i+2) by arranging points on a concentric circle and choosing i points uniformly. Any disk that contains more than this fraction must intersect one of the selected points. A complementary lower bound is constructed by clustering points near the centre and spreading the chosen i points on the periphery, showing that ε_i^𝔻 = Θ(1/i). While the exact constant is not pinned down, the asymptotic order matches the upper bound, indicating that strong ε‑nets for disks shrink only linearly with i, unlike the exponential decay for rectangles.

The paper situates these results within the broader landscape of ε‑net theory. Classical ε‑nets guarantee a net of size O(1/ε·log(1/ε)) for arbitrary range families with finite VC‑dimension. In contrast, the strong ε‑nets studied here achieve dramatically smaller sizes for specific, highly structured families: exponential decay for axis‑parallel rectangles and linear decay for halfspaces and disks. This demonstrates that when the range family possesses strong combinatorial regularities (e.g., product structure of boxes), far more compact summaries of the data are possible.

Beyond the technical contributions, the authors discuss several open problems. Determining tight strong ε‑net bounds for higher‑dimensional halfspaces or for other non‑orthogonal families remains unresolved. For disks, narrowing the gap between the known upper and lower constants is an explicit challenge. Moreover, the notion of “approximate strong centerpoints” or “partial strong ε‑nets” for families where exact strong centerpoints do not exist is proposed as a promising direction for future work.

In summary, the paper makes three core advances: (1) it proves the existence of a strong centerpoint for axis‑parallel boxes with an optimal fraction 1/2^d; (2) it derives exact strong ε‑net values ε_i^ℛ = 1/2^{i+1} for rectangles, ε_i^ℋ = 1/(i+1) for halfspaces, and Θ(1/i) for disks in the plane; and (3) it highlights how these specialized strong ε‑nets dramatically improve over generic ε‑net bounds for certain geometric range families. These findings deepen our understanding of geometric data summarization and have potential applications in range‑query data structures, clustering, and approximation algorithms where compact, provably representative subsets are essential.