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Toussaint [8] introduced the sphere-of-influence graph of a finite set of points in Euclidean space for applications in pattern analysis and image processing (see [7] for a recent survey). This notion was later generalized to so-called closed sphere-of-influence graphs [3] and to k-th closed sphere-of-influence graphs [4]. Our setting will be a d-dimensional normed space \(\mathcal N\) with norm \(\left||\cdot \right||\). We denote the ball with center \(c\in \mathcal N\) and radius r by B(c, r).

FormalPara Definition 1

Let \(k\in \mathbb N\) and let \(V=\{c_i:i=1,\dots ,m\}\) be a set of points in the d-dimensional normed space \(\mathcal N\). For each \(i\in \{1,\dots ,m\}\), let \(r_i^{(k)}\) be the smallest r such that

$$ \{j\in \mathbb N:j\ne i, \left||c_i-c_j\right||\le r\} $$

has at least k elements. Define the k-th closed sphere-of-influence graph on V by setting \(\{c_i, c_j\}\) an edge whenever \(B(c_i,r_i^{(k)})\cap B(c_j,r_j^{(k)})\ne \emptyset \).

Füredi and Loeb [1] gave an upper bound for the minimum degree of any closed sphere-of-influence graph in \(\mathcal N\) in terms of a certain packing quantity of the space (see also [5, 6].)

FormalPara Definition 2

Let \(\vartheta (\mathcal N)\) denote the largest cardinality of a subset A of the ball B(o, 2) of the normed space \(\mathcal N\) such that any two points of A are at distance at least 1, and the origin o is in A.

Füredi and Loeb [1] showed that any closed sphere-of-influence graph (that is, in our terminology, a first closed sphere-of-influence graph) in \(\mathcal N\) has a vertex of degree smaller than \(\vartheta (\mathcal N)\le 5^d\). (It is clear that \(\vartheta (\mathcal N)\) is bounded above by the number of balls of radius 1 / 2 that can be packed into a ball of radius 5 / 2, which is at most \(5^d\) by volume considerations.)

Guibas, Pach and Sharir [2] showed that any k-th closed sphere-of-influence graph in d-dimensional Euclidean space has a vertex of degree at most \(c^dk\), for some universal constant \(c>1\). In this note we show the following more precise result, valid for all norms, and generalizing the result of Füredi and Loeb [1] mentioned above.

FormalPara Theorem 3

Every k-th sphere-of-influence graph on at least two points in a normed space \(\mathcal N\) has at least two vertices of degree smaller than \(\vartheta (\mathcal N)k\le 5^d k\).

We note that the theorem still holds when there are repeated elements.

FormalPara Corollary 4

A k-th sphere-of-influence graph on n points in \(\mathcal N\) has at most \((\vartheta (\mathcal N)k-1)n\le (5^d k -1)n\) edges.

Proof of Theorem 3 Let \(V=\{c_1,c_2,\dots ,c_m\}\). Relabel the vertices \(c_1,c_2,\dots ,c_m\) such that \(r_1^{(k)}\le r_2^{(k)}\le \dots \le r_m^{(k)}\). We define an auxiliary graph H on V by joining \(c_i\) and \(c_j\) whenever \(\left||c_i-c_j\right||<\max \{r_i^{(k)},r_j^{(k)}\}\). Thus, if \(\{c_i:i\in I\}\) is an independent set in H, then no ball in \(\{B(c_i,r_i^{(k)}):i\in I\}\) contains the center of another in its interior. We next bound the chromatic number of H.

FormalPara Lemma 5

The chromatic number of H does not exceed k.

FormalPara Proof

Note that for each \(i\in \{1,\dots ,m\}\), the set

$$\{j<i:c_ic_j\in E(H)\} = \{j<i:\left||c_i-c_j\right||<r_i^{(k)}\}$$

has less than k elements. Therefore, we can greedily color H in the order \(c_1, c_2,\dots , c_m\) by k colors.    \(\square \)

We next show that the degrees of \(c_1\) and \(c_2\) (corresponding to the two smallest values of \(r_i^{(k)}\)) are both at most \(\vartheta (\mathcal N)k\), which will complete the proof of Theorem 3. We first need the so-called “bow-and-arrow” inequality of [1]. For completeness, we include the proof from [1].

FormalPara Lemma 6

(Füredi–Loeb [1]) For any two non-zero elements a and b of a normed space,

$$\begin{aligned} \left||\frac{1}{\left||a\right||}a-\frac{1}{\left||b\right||}b\right||\ge \frac{\left||a-b\right||-\left|\left||a\right||-\left||b\right||\right|}{\left||b\right||}. \end{aligned}$$
FormalPara Proof

Without loss of generality, we may assume that \(\left||a\right||\ge \left||b\right||>0\). Then

$$\begin{aligned} \left||a-b\right||&=\left||\left||a\right||\frac{1}{\left||a\right||}a-\left||b\right||\frac{1}{\left||b\right||}b\right||\\&=\left||\left||b\right||(\frac{1}{\left||a\right||}a-\frac{1}{\left||b\right||}b)+(\left||a\right||-\left||b\right||)\frac{1}{\left||a\right||}a\right||\\&\le \left||b\right||\left||\frac{1}{\left||a\right||}a-\frac{1}{\left||b\right||}b\right||+\left||a\right||-\left||b\right||. \square \end{aligned}$$

The next lemma is abstracted with minimal hypotheses from [5, Proof of Theorem 6] (see also [1, Proof of Theorem 2.1]).

FormalPara Lemma 7

Consider the balls \(B(v_1,\lambda _1)\) and \(B(v_2,\lambda _2)\) in the normed space \(\mathcal N\), such that \(\max \{\lambda _1,\lambda _2\}\ge 1\), \(v_1\notin {\text {int}}(B(v_2,\lambda _2))\), \(v_2\notin {\text {int}}(B(v_1,\lambda _1))\) and \(B(v_i,\lambda _i)\cap B(o,1)\ne \emptyset \) (\(i=1,2\)). Define \(\pi :\mathcal N\rightarrow B(o,2)\) by

$$ \pi (x)={\left\{ \begin{array}{ll} x &{}\,\text {if}\,\left||x\right||\le 2,\\ \frac{2}{\left||x\right||}x &{} \,\text {if}\,\left||x\right||\ge 2. \end{array}\right. } $$

Then \(\left||\pi (v_1)-\pi (v_2)\right||\ge 1\).

FormalPara Proof

In terms of the norm, we are given that \(\left||v_1-v_2\right||\ge \max \{\lambda _1,\lambda _2\}\ge 1\), \(\left||v_1\right||\le \lambda _1+1\), and \(\left||v_2\right||\le \lambda _2+1\). Without loss of generality, we may assume that \(\left||v_2\right||\le \left||v_1\right||\).

If \(v_1,v_2\in B(o,2)\) then \(\left||\pi (v_1)-\pi (v_2)\right||=\left||v_1-v_2\right||\ge 1\).

If \(v_1\notin B(o,2)\) and \(v_2\in B(o,2)\), then

$$\begin{aligned} \left||\pi (v_1)-\pi (v_2)\right||&=\left||2\frac{1}{\left||v_1\right||}v_1-v_2\right|| \ge \left||v_1-v_2\right||-\left||v_1-2\frac{1}{\left||v_1\right||}v_1\right||\\&= \left||v_1-v_2\right||-(\left||v_1\right||-2)\ge \lambda _1 - (\lambda _1+1) + 2 =1. \end{aligned}$$

If \(v_1,v_2\notin B(o,2)\), then

$$\begin{aligned} \left||\pi (v_1)-\pi (v_2)\right||&=\left||2\frac{1}{\left||v_1\right||}v_1-2\frac{1}{\left||v_2\right||}v_2\right|| \ge 2\frac{\left||v_1-v_2\right||-\left||v_1\right||+\left||v_2\right||}{\left||v_2\right||} \quad \text {by Lemma~6}\\&\ge 2\left( \frac{\lambda _1-(\lambda _1 + 1)}{\left||v_2\right||} + 1\right) = \frac{-2}{\left||v_2\right||} + 2 \ge -1+2=1. \quad \quad \quad \quad \quad \quad \quad \quad \square \end{aligned}$$

We can now finish the proof of Theorem 3. Let \(i\in \{1,2\}\), and let \(c:=c_i\), that is, the radius corresponding to c is the smallest, or second smallest. By Lemma 5 we can partition the set of neighbors of c in the k-th closed sphere-of-influence graph on V into k classes \(N_1,\dots ,N_k\) so that each \(N_t\) is an independent set in H. We may assume that the radius \(r_i^{(k)}\) corresponding to c is 1. Then for any \(t\in \{1,\dots ,k\}\), each ball in \(\{B(c_j,r_j^{(k)}):c_j\in N_t\}\) intersects B(c, 1), and the center of no ball is in the interior of another ball. By Lemma 7, \(\{\pi (p-c):p\in N_t\}\) is a set of points contained in B(o, 2) with a distance of at least 1 between any two. That is, \(\left|N_t\setminus {\text {int}}(B(c,1))\right| \le \vartheta (\mathcal N)-1\) for each \(t=1,\dots ,k\). Since there are at most \(k-1\) points in \(V\cap {\text {int}}(B(c,1))\setminus \{c\}\), it follows that the degree of c is at most \(\sum _{t=1}^k \left|N_t\setminus {\text {int}}(B(c,1))\right| + k-1 \le (\vartheta (\mathcal N)-1)k + k-1 = \vartheta (\mathcal N)k-1\).