## 4 Convex Minimization Problem
If the Step 6 inequality holds for each node, then each point of the ball centered at \(\dfrac{\mathbf{v}+\mathbf{u}_{j}}{2}\) with radius \(\left|\dfrac{\mathbf{v}-\mathbf{u}_{j}}{2}\right|_{2}\) is contained in the \(l_{2}\) ball of radius \(\delta\) centered at \(\mathbf{v}\) (see Figure 2). Hence the sufficient condition offered by Algorithm 4.1 is_ **more** _conservative than the one suggested in [8]._Algorithm 4.1 can be executed with a variety of different norms, and, as we show next, \(l_{2}\) might not be the best one when communication between the nodes should be minimized. **Remark 4.2**: _Let \(n=d=2\),_  
\[f(\mathbf{x})=|x_{1}-1|+|x_{2}-1|=\|\mathbf{x}-\mathbf{e}\|_{1}\]  
_the distance is given by the \(l_{1}\) norm, and the aim is to monitor the inequality \(f(\mathbf{v})-1>0\). Let_  
\[\mathbf{v}_{1}(t_{0})=\left[ \begin{array}{c}1\\ 0\end{array} \right],\ \mathbf{v}_{2}(t_{0})=\left[ \begin{array}{c}-1\\ 0\end{array} \right]\]  
\[\mathbf{v}_{1}(t_{1})=\left[ \begin{array}{c}1.9\\ 0\end{array} \right],\ \mathbf{v}_{2}(t_{1})=\left[ \begin{array}{c}-1\\ 0\end{array} \right]\]  
_We first consider the \"ball cover\" construction suggested in [8]. With this data \(\mathbf{v}(t_{0})=0\) with \(f(\mathbf{v}(t_{0}))=2\), and \(\mathbf{v}(t_{1})=\left[ \begin{array}{c}0.45\\ 0\end{array} \right]\) with \(f(\mathbf{v}(t_{1}))=1.55\). At the same time \(\mathbf{u}_{1}(t_{1})=\mathbf{v}(t_{0})+\left[\mathbf{v}_{1}(t_{1})-\mathbf{ v}_{1}(t_{0})\right]=\left[ \begin{array}{c}0.9\\ 0\end{array} \right]\). It is easy to see that the \(l_{2}\) ball of radius \(\left\|\frac{\mathbf{v}(t_{0})-\mathbf{u}_{1}(t_{1})}{2}\right\|_{2}\) centered at \(\frac{\mathbf{v}(t_{0})+\mathbf{u}_{1}(t_{1})}{2}\) intersects the \(l_{1}\) ball of radius \(1\) centered at \(\left[ \begin{array}{c}1\\ 1\end{array} \right]\) (see Figure 3). Hence the algorithm suggested in [8] requires nodes to communicate at time \(t_{1}\)._  
Figure 2: conservative cover by a single \(l_{2}\) ball. _On the other hand the \(l_{1}\) distance from \(\mathbf{v}(t_{0})\) to the set \(\{\mathbf{x}\ :\ \|\mathbf{x}-\mathbf{e}\|_{1}=1\}\) is \(1\), and since_  
\[\|\mathbf{v}_{1}(t_{1})-\mathbf{v}_{1}(t_{0})\|_{1}<1,\text{ and }\|\mathbf{v}_{2} (t_{1})-\mathbf{v}_{2}(t_{0})\|_{1}<1\]  
_Algorithm 4.1 requires no communication between nodes at time \(t_{1}\). In this particular case the sufficient condition offered by Algorithm 4.1 is_ **less** _conservative than the one suggested in [8]._  
**Remark 4.3**: _It is easy to see that inclusion Equation (8) fails when \(B_{1}(\mathbf{x},\mathbf{y})\) is an \(l_{1}\) ball of radius \(\dfrac{1}{2}\|\mathbf{x}-\mathbf{y}\|_{1}\) centered at \(\dfrac{\mathbf{x}+\mathbf{y}}{2}\). Indeed, when, for example,_  
\[\mathbf{v}=\left[ \begin{array}{c}0\\ 0.5\end{array} \right],\ \mathbf{u}_{1}=\left[ \begin{array}{c}-1\\ 0\end{array} \right],\ \mathbf{u}_{2}=\left[ \begin{array}{c}1\\ 0\end{array} \right]\]  
_(see Figure 4) one has_  
\[\mathrm{conv}\ \{\mathbf{v},\mathbf{u}_{1},\mathbf{u}_{2}\}\
ot\subset B_{1}( \mathbf{v},\mathbf{u}_{1})\cup B_{1}(\mathbf{v},\mathbf{u}_{2})\]  
_In the next section we apply Algorithm 4.1 to a real life data and report number of required mean computations._  
Figure 3: \(l_{2}\) ball cover requires communication.