# A Numerical Approach for the Filtered Generalized Cech Complex
## 2 Vietoris-Rips and Cech systems
: _Figure 1 shows (left picture) the following disk system in \(\mathbb{R}^{2}\),_  
\[M=\{D_{1}((-3,0);4),D_{2}((1,3);3),D_{3}((2,-1);2)\}.\]  
_This 2-disk system is a Vietoris-Rips system and also a Cech system. In this case, we have \(\
u_{M}=\sqrt{26}/6\approx 0.8498\), and, in actuality, the \(\
u_{M}\)-rescaled 2-disk system \(M_{\
u_{M}}\) (center picture) has an empty intersection, i.e., \(\bigcap_{D_{i}\in M}D_{i}(c_{i};\
u_{M}\,r_{i})=\emptyset\), so it corresponds to a Vietoris Rips system which is not a Cech system. For such 2-disk system, the Cech.scale algorithm in Section 5 yields to \(\mu_{M}=0.9188\) and the Cech system \(M_{\mu_{M}}\) is shown in the right picture._  
There exists a close relationship between Vietoris-Rips systems and Cech systems. Obviously, every Cech system is also a Vietoris-Rips system, but the opposite statement does not hold in general (as we saw in the example above). However, we have the following result which extends the standard Vietoris-Rips Lemma ([17], Theorem 2.5), to any Vietoris-Rips system; such result can be found in ([18], Theorem 3.2) in the context of weighted simplicial complexes. The standard Vietoris-Rips Lemma is established in the context of a disk system in which all radii are equal, and corresponds to a reformulation of the well-known [PERSON]'s Lemma [19]. On the other hand, for a disk system in general, the Vietoris-Rips Lemma is also valid, although it does not follow directly from [PERSON]'s Lemma. Next, we propose a proof of the Vietoris-Rips Lemma, for an arbitrary disk system, using elementary geometrical arguments. **Lemma 2**. : _Let \(M=\{D_{i}(c_{i};r_{i})\subset\mathbb{R}^{d}\mid r_{i}>0\}\) be a finite set of closed disks in \(\mathbb{R}^{d}\). If \(D_{i}(c_{i};r_{i})\cap D_{j}(c_{j};r_{j})\
eq\emptyset\) for every pair of disks in \(M\), then_  
\[\bigcap_{D_{i}\in M}D_{i}(c_{i};\sqrt{2d/(d+1)}\,r_{i})\
eq\emptyset.\]  
In other words, for every Vietoris-Rips system \(M\) in \(\mathbb{R}^{d}\), the \(\sqrt{2d/(d+1)}\)-rescaled disk system \(M_{\sqrt{2d/(d+1)}}\) is a Cech system. Proof. : First, we prove the result for the case where \(M\) has at most \(d+1\) disks, say \(M=\{D_{1},\ldots,D_{d^{\prime}}\}\), \(d^{\prime}\leq d+1\). Note that we need to prove that \(\mu_{M}\leq\sqrt{\frac{2d}{d+1}}\). Let \(\{c_{M}\}=\bigcap_{i}D_{i}(c_{i};\mu_{M}r_{i})\) be the unique intersection point of the \(\mu_{M}\)-rescaled disk system (see Lemma 1). Without loss of generality, we assume that \(\|c_{i}-c_{M}\|=\mu_{M}r_{i}\) for \(c_{1},\ldots,c_{m}\) and \(m\leq d^{\prime}\). Then, \(c_{M}\) belongs to the convex hull of the set \(\{c_{1},\ldots,c_{m}\}\), for if this were not true, there would exist an hyperplane across \(c_{M}\) such that the set \(\{c_{1},\ldots,c_{m}\}\) is completely contained at one side, let  
Figure 1: Vietoris–Rips and Čech systems in the plane. be the normal vector for such hyperplane in the opposite direction, then \(\langle v,c_{i}-c_{M}\rangle>0\) for all \(i=1,\ldots,m\), therefore  
\[\|c_{i}-c_{M}\|^{2}=\|c_{i}-(c_{M}+tv)\|^{2}+2t\langle v,c_{i}-c_{M}\rangle-t^{ 2}\|v\|^{2}>\|c_{i}-(c_{M}+tv)\|^{2}\]  
for every \(t\in I_{v}:=(0,2\langle v,c_{i}-c_{M}\rangle/\|v\|^{2})\); this implies that \(c_{M}+tv\in D(c_{i};\mu_{M}r_{i})\) for any \(i=1,\ldots,m\) and \(t\in I_{v}\), which is a contradiction since \(c_{M}\) is the only point in the intersection \(\bigcap_{i}D(c_{i};\mu_{M}r_{i})\). Therefore, \(c_{M}\) is in the convex hull. Now, define \(\hat{c}_{i}:=c_{i}-c_{M}\) and let \(\theta_{ij}\) denote the angle between vectors \(\hat{c}_{i}\) and \(\hat{c}_{j}\). As \(c_{M}\) is in the convex hull of \(\{c_{1},\ldots,c_{m}\}\), then the vector \(0\in\mathbb{R}^{d}\) can be written as a convex combination \(\sum_{j=1}^{m}a_{j}\hat{c}_{j}=0\).