---
title: 'Strong ε-Net: Bounds & Constructions'
url: https://www.emergentmind.com/topics/strong-net
type: topic
---

# Strong ε-Net: Bounds & Constructions

A strong $\epsilon$-net is a combinatorial geometric object that provides a highly constrained representative subset of a finite point set in $\mathbb{R}^d$ with respect to a specified family of geometric ranges. For a given finite set $P$ of $n$ points and a family $\mathcal{C}$ of (typically convex) subsets of $\mathbb{R}^d$, a strong $\epsilon$-net is a subset $N \subseteq P$ such that every range $C \in \mathcal{C}$ containing more than $\epsilon n$ points of $P$ also contains at least one point of $N$. The defining property of a strong $\epsilon$-net, in contrast with weak $\epsilon$-nets, is that its constituent elements are required to be points of $P$ itself rather than arbitrary points in the ambient space. The article "Small Strong Epsilon Nets" [1208.2785] introduces and analyzes *small strong $\epsilon$-nets*, providing exact and nearly tight upper and lower bounds on their minimal size and covering fraction for key geometric range families in the plane and higher dimensions.

## 1. Formal Definitions

Let $P \subset \mathbb{R}^d$ be a finite set of $n$ points and $\mathcal{C}$ a family of geometric ranges. The central notions are:

- **Strong $\epsilon$-net**: A subset $N \subseteq P$ is a strong $\epsilon$-net for $(P, \mathcal{C})$ if for every $C\in \mathcal{C}$ with $|C \cap P| > \epsilon n$, the intersection $N \cap C$ is nonempty.
- **Weak $\epsilon$-net**: As above, but $N$ need only be a subset of $\mathbb{R}^d$, not necessarily of $P$.
- **Centerpoint and Strong Centerpoint**: A centerpoint $x \in \mathbb{R}^d$ of $P$ is a point lying in every convex set containing more than $dn/(d+1)$ points of $P$; a strong centerpoint for $\mathcal{C}$ is $p \in P$ contained in every $C \in \mathcal{C}$ with $|C \cap P| > \alpha n$ for some constant $\alpha < 1$. In this language, a strong centerpoint is a one-point strong $\epsilon$-net.
- **Small strong $\epsilon$-nets**: For a family $\mathcal{S}$ of ranges, define $\epsilon_i^{\mathcal{S}}$ as the smallest $\epsilon \in [0,1]$ such that every $n$-point set $P \subset \mathbb{R}^d$ admits a strong $\epsilon$-net $N \subseteq P$ of size $i$, i.e., $|N| = i$.

These definitions set the stage for the quantitative analysis of how well a finite subset of $P$ can represent all large-range subsets defined by $\mathcal{C}$.

## 2. Main Results for Core Geometric Range Families

Three principal families are examined in the plane: axis-parallel rectangles ($\mathcal{R}$), halfspaces ($\mathcal{H}$), and disks ($\mathcal{D}$). The paper provides explicit finite-size bounds and asymptotic rates for $\epsilon_i^{\mathcal{S}}$ for each:

### (I) Axis-parallel Rectangles $(\mathcal{R})$

- **Strong Centerpoint**: For axis-parallel boxes in $\mathbb{R}^d$, $\epsilon_1^{\mathcal{R}} = (2d-1)/(2d)$. In $\mathbb{R}^2$, this specializes to $\epsilon_1^{\mathcal{R}} = 3/4$.
- **Small Nets in $\mathbb{R}^2$**: For $1 \leq i \leq 10$, explicit lower (LB) and upper (UB) bounds for $\epsilon_i^{\mathcal{R}}$ are as follows:

| $i$ | 1      | 2     | 3     | 4    | 5     | 6     | 7      | 8     | 9      | 10    |
|-----|--------|-------|-------|------|-------|-------|--------|-------|--------|-------|
| LB  | $3/4$  | $5/9$ | $2/5$ | $3/10$ | $1/4$ | $1/5$ | $5/29$ | $2/13$ | $3/22$ | $1/8$|
| UB  | $3/4$  | $5/8$ | $9/16$| $1/2$| $15/32$| $15/32$| $3/7$ | $2/5$ | $5/13$ | $3/8$|

- **Asymptotic Bounds**: $\epsilon_i^{\mathcal{R}} \ge 10/(9i)$ for all $i \ge 2$; $\epsilon_i^{\mathcal{R}} = O(1/i)$ by recursive grid and slicing constructions.

### (II) Halfspaces $(\mathcal{H})$ in $\mathbb{R}^2$

- **No Strong Centerpoint**: $\epsilon_1^{\mathcal{H}} = 1$.
- **Upper Bounds**: $\epsilon_i^{\mathcal{H}} \le 2/(i+1)$ for all $i \ge 1$.
- **Lower Bounds**: For odd $i$, $\epsilon_i^{\mathcal{H}} = 2/(i+1)$ (tight); for even $i$, $2/(i+2) \le \epsilon_i^{\mathcal{H}} \le 2/(i+1)$. Notably, $\epsilon_2^{\mathcal{H}} \in [3/5, 2/3]$, $\epsilon_3^{\mathcal{H}} = 1/2$.

### (III) Disks $(\mathcal{D})$ in $\mathbb{R}^2$

- **No Strong Centerpoint**: $\epsilon_1^{\mathcal{D}} = 1$.
- **Upper Bound (for $i=2$)**: $\epsilon_2^{\mathcal{D}} \le 2/3$.
- **Lower Bounds**: For odd $i$, $\epsilon_i^{\mathcal{D}} \ge 2/(i+1)$; for even $i$, $\epsilon_i^{\mathcal{D}} \ge 2/(i+2)$. For $i=3$, there is an explicit construction yielding $\epsilon_3^{\mathcal{D}} \ge 1/3$.

## 3. Fundamental Proof Techniques

Several foundational methodologies underpin the analysis and construction of small strong $\epsilon$-nets:

- **Slicing Arguments (Rectangles)**: Partition $P$ using one or two coordinate-aligned slabs to localize high-density regions and recurse, yielding moderate-sized nets.
- **Grid Arguments (Theorem 3.1)**: Partition $P$ into $x$ horizontal and $y$ vertical strips; select near-boundary neighbors to block empty rectangles, then apply recursion.
- **Convex Hull "Necklace" (Halfspaces)**: Traverse the convex hull of $P$, selecting hull vertices so that each large exterior halfspace must contain a selected point.
- **Delaunay/Empty-Circle Argument (Disks)**: Use the Delaunay triangulation around a centerpoint and the crossing count of segments through triangle edges to force that large disks intersect certain key points.
- **Recursive Lower Bound**: Construct two well-separated hard instances and combine, showing $\epsilon_{j+k}^{\mathcal{S}} \ge (\epsilon_j^{\mathcal{S}} \epsilon_k^{\mathcal{S}}) / (\epsilon_j^{\mathcal{S}} + \epsilon_k^{\mathcal{S}})$ for any convex family $\mathcal{S}$ and $j,k \ge 1$.

## 4. Explicit Constructions

Theoretical bounds are realized via concrete algorithmic constructions for small strong $\epsilon$-nets:

- **Axis-parallel Rectangles**: Starting from a strong centerpoint $q$ (with covering ratio $3/4$), draw vertical and horizontal lines through $q$, partitioning $P$ into four slabs. If any slab contains a large fraction, place a net there and its complement; if not, recursions on two slabs of half-size produce higher-order nets while optimizing the balance parameter $\delta$.
- **Halfspaces**: Maintain a set starting at a hull vertex, walk along the hull, and, upon encountering a halfspace that would avoid all previously selected points and contain too many points, add the current vertex. Ensure at most $i$ points are selected.
- **Disks**: Compute a centerpoint $p$ and the Delaunay triangulation. The triangle containing $p$ (say, $abc$) provides an edge (e.g., $ab$) heavily crossed by centerpoint-complement segments. The points $a$ and $b$ form the net, as any disk avoiding them can only capture a limited fraction of $P$.

## 5. Tabulation of Bounds and Asymptotics

Key ranges and their small-$i$ behaviors in the plane are summarized:

| $i$              | $\epsilon_i^{\mathcal{R}}$         | $\epsilon_i^{\mathcal{H}}$               | $\epsilon_i^{\mathcal{D}}$                   |
|------------------|------------------------------------|-------------------------------------------|----------------------------------------------|
| $1$              | $3/4$                              | $1$                                      | $1$                                         |
| $2$              | $[5/9, 5/8]$                       | $[3/5, 2/3]$                             | $[3/5, 2/3]$                                |
| $3$              | $[2/5, 9/16]$                      | $1/2$                                    | $[1/3, 2/3]$                                |
| Asymptotics      | $\epsilon_i^{\mathcal{R}}\sim O(1/i)$| $2/(i+1)$ (odd $i$), $[2/(i+2), 2/(i+1)]$ even $i$ | $\ge 2/(i+2)$, upper bounds similar to halfspaces (except $i=2$) |

These results yield the first nontrivial upper and lower bounds for constant-size strong $\epsilon$-nets for the considered range families.

## 6. Open Problems and Research Directions

Several directions remain open for exploration:

- Determining the *exact* values of $\epsilon_i^{\mathcal{R}}$ for $i=2,3$, $\epsilon_i^{\mathcal{H}}$ for even $i$, and $\epsilon_i^{\mathcal{D}}$ for $i\ge 3$.
- Improving upper bounds for disks beyond $i=2$ and closing the gap between bounds of type $2/(i+1)$ (from halfspace-style constructions) and the currently known lower bounds.
- Extending strong $\epsilon$-net bounds to other low-VC-dimension families such as pseudo-disks or convex $k$-gons [1208.2785].

The study of small strong $\epsilon$-nets delineates the precise interaction between geometric range families and discrete point set coverage, distinguishing itself from the theory of weak $\epsilon$-nets by the requirement that covering points lie in $P$. This work establishes existence results for strong centerpoints of axis-parallel boxes in arbitrary dimension and opens avenues for understanding combinatorial geometry through the lens of point set representativity and geometric range complexity.

Source: https://www.emergentmind.com/topics/strong-net