## 3 Non-Linear Threshold Function: An Example  
We start with a slight modification of a simple one dimensional example presented in [8]. **Example 3.1**: _Let \(f(x)=x^{2}-9\), and \(v_{i}\), \(i=1,2\) are scalar values stored at two distinct nodes. Note that if \(v_{1}=-4\), and \(v_{2}=4\), then_  
\[f(v_{1})=f(v_{2})=7>0\text{ and }\]  
\[f\left(\frac{v_{1}+v_{2}}{2}\right)=-9<0\]  
_If \(v_{1}=-2\), and \(v_{2}=6\), then_  
\[f(v_{1})=-5<0,\ f(v_{2})=27>0\text{ and }\]  
\[f\left(\frac{v_{1}+v_{2}}{2}\right)=-5<0\]  
_Finally, when \(v_{1}=2\), and \(v_{2}=6\) one has_  
\[ \begin{array}{c}f(v_{1})=-5<0,\ f(v_{2})=27>0\\ \text{ and }\\ f\left(\frac{v_{1}+v_{2}}{2}\right)=7>0\end{array}  \tag{7}\]  
The simple illustrative example leads the authors of [8] to conclude that it is impossible to determine from the values of \(f\) at the nodes whether its value at the average is above the threshold or not. The remedy proposed is to consider the vectors \(\mathbf{u}_{j}(t)=\mathbf{v}(t_{i})+[\mathbf{v}_{j}(t)-\mathbf{v}_{j}(t_{i})]\), \(j=1,\ldots,n\), \(t\geq t_{i}\) and to monitor the values of \(f\) on the convex hull \(\mathrm{conv}\ \{\mathbf{u}_{1}(t),\ldots,\mathbf{u}_{n}(t)\}\) instead of the value of \(f\) at the average Equation (1). This strategy leads to sufficient conditions for Equation (1), and may be conservative. The monitoring techniques for values of \(f\) on \(\mathrm{conv}\ \{\mathbf{u}_{1}(t),\ldots,\mathbf{u}_{n}(t)\}\) without communication between the nodes are based on the following two observations:  
1. _Convexity property_. The mean \(\mathbf{v}(t)\) is given by \(\frac{\mathbf{v}_{1}(t)+\ldots+\mathbf{v}_{n}(t)}{n}=\frac{\mathbf{u}_{1}(t) +\ldots+\mathbf{u}_{n}(t)}{n}\), _i.e._, the mean \(\mathbf{v}(t)\) is in the convex hull of \(\{\mathbf{u}_{1}(t),\ldots,\mathbf{u}_{n}(t)\}\), and \(\mathbf{u}_{j}(t)\) is available to node \(j\) without much communication with other nodes. 2. If \(B_{2}(\mathbf{x},\mathbf{y})\) is an \(l_{2}\) ball of radius \(\frac{1}{2}\|\mathbf{x}-\mathbf{y}\|_{2}\) centered at \(\frac{\mathbf{x}+\mathbf{y}}{2}\), then \[\mathrm{conv}\ \{\mathbf{v},\mathbf{u}_{1},\ldots,\mathbf{u}_{n}\}\subseteq \bigcup_{j=1}^{n}B_{2}(\mathbf{v},\mathbf{u}_{j})\] (8)  
(see Figure 1). Since each ball  
\[B_{2}(\mathbf{v}(t_{i}),\mathbf{u}_{j}(t)),\ t\geq t_{i},\ j=1,\ldots,n \tag{9}\]  
can be monitored by node \(j\) with no communication with other nodes, Equation (8) allows to split monitoring of \(\mathrm{conv}\ \{\mathbf{v}(t_{i}),\mathbf{u}_{1}(t),\ldots,\mathbf{u}_{n}(t)\}\), \(t\geq t_{i}\) into \(n\) independent tasks executed by the \(n\) nodes separately and without communication. While the inclusion Equation (8) holds when \(B_{2}\) is substituted by \(B_{p}\) with \(p\geq 2\) as we show later (see Remark 4.3) the inclusion fails when, for example, \(p=1\) (for experimental results obtained with different norms see Section 5). In this paper we propose an alternative strategy that will be briefly explained next using Example 3.1, \(f(x)=x^{2}-9\), and assignment provided by Equation (7). Let \(\delta\) be a positive number. Consider two intervals of radius \(\delta\) centered at \(v_{1}=2\) and \(v_{2}=6\), _i.e._, we are interested in the intervals  
\[[2-\delta,2+\delta],\text{ and }[6-\delta,6+\delta]\]  
If \(v_{1}(t)\in[2-\delta,2+\delta]\), \(v_{2}(t)\in[6-\delta,6+\delta]\), and \(\delta\) is small, then the average \(\dfrac{v_{1}(t)+v_{2}(t)}{2}\) is not far from \(\dfrac{2+6}{2}\), and \(f\left(\dfrac{v_{1}(t)+v_{2}(t)}{2}\right)\) is not far from \(7\) (hence positive). In fact the sum of the intervals is the interval \([8-2\delta,8+2\delta]\), and  
\[4-\delta\leq\dfrac{v_{1}(t)+v_{2}(t)}{2}\leq 4+\delta\]  
The \"zero\" points \(Z_{f}\) of \(f\) are \(-3\) and \(3\), and as soon as \(\delta\) is large enough so that the interval \([4-\delta,4+\delta]\) \"hits\" a point where \(f\) vanishes, communication between the nodes is required in order to verify Equation (1).