## 4 Proof of Guarantees  
Let \(M\) denote the output mesh of DelPSC.  
**Theorem 2**: \(M\) _has guarantee_ G1_._  
At the end of Mesh2 Complex the disk condition ensures that \(\operatorname{Skl}^{2}S|_{\sigma}\) is a simplicial complex where each vertex \(v\) belongs to \(\sigma\) and has a \(2\)-disk as its star. It follows from a result in PL topology that \(\operatorname{Skl}^{2}S|_{\sigma}\) is a \(2\)-manifold when DelPSC terminates.  
The boundary of \(\operatorname{Skl}^{2}S|_{\sigma}\) has all weighted vertices in \(\operatorname{bd}\sigma\). Each such point \(p\) is connected to its adjacent vertices in \(\operatorname{bd}\sigma\) by the disk condition. Therefore, the boundary of \(\operatorname{Skl}^{2}S|_{\sigma}\) consists of edges that connect adjacent vertices in \(\operatorname{bd}\sigma\) and hence this boundary is homeomorphic to \(\operatorname{bd}\sigma\).  
To prove G2 we use a result of [PERSON] and [PERSON] [17] about the extended topological ball property (TBP). It can be shown that the following two properties P1 and P2 imply the extended TBP [8]. Therefore, according to the Edelsbrunner-Shah [17] result, the underlying space of \(\operatorname{Del}S|_{\mathcal{D}}\) is homeomorphic to the \(|\mathcal{D}|\) if P1 and P2 hold. Let \(F\) be a \(k\)-face of \(\operatorname{Vor}S\) where \(S\) is the output vertex set.  
* If \(F\) intersects an element \(\sigma\in\mathcal{D}_{j}\subseteq\mathcal{D}\), the intersection is a closed \((k+j-3)\)-ball.
* There is a unique lowest dimensional element \(\sigma_{F}\in\mathcal{D}\) so that \(F\) intersects \(\sigma_{F}\) and only elements that are incident to \(\sigma_{F}\).  
Lemma 2 almost provides condition P1 except for the case that \(V_{p}\) may intersect a patch \(\tau\) where \(p\
ot\in\tau\) (Figure 7 (middle, right)). Lemma 8 establishes that this is not possible. Lemma 9 gives P2. Proofs of both of them appear in the appendix.  
**Lemma 8**: _There exists a constant \(\lambda_{c}>0\) so that following holds. Let \(S\) be the point set output by DelPSC\((\mathcal{D},\lambda)\) for some \(\lambda\leqslant\lambda_{c}\). Then for each point \(p\in S\), \(V_{p}|_{\mathcal{D}}=\bigcup_{\sigma\
i p}\sigma_{p}\)._  
**Lemma 9**: _Let \(S\) be the point set as defined in Lemma 8. Let \(F\) be a \(k\)-face in \(\operatorname{Vor}S\). There is an element \(\sigma_{F}\in\mathcal{D}\) so that \(F\) intersects \(\sigma_{F}\) and only elements in \(\mathcal{D}\) that have \(\sigma_{F}\) on their boundary._  
**Theorem 3**: \(M\) _satisfies_ G2_._  
For a sufficiently small \(\lambda>0\), DelPSC satisfies the conditions of Lemma 8 and Lemma 9. This means that properties P1 and P2 are satisfied when \(\lambda\) is sufficiently small. Also when P1 and P2 are satisfied \(\bigcup_{i}\operatorname{Skl}^{i}S|_{\mathcal{D}}=\operatorname{Del}S|_{ \mathcal{D}}\). It follows that the Edelsbrunner-Shah conditions are satisfied for the output \(M\) of DelPSC. Thus, \(M\) has an underlying space homeomorphic to \(|\mathcal{D}|\). The homeomorphism constructed by [PERSON] and [PERSON] actually respects the stratification, that is, for each \(\sigma\in\mathcal{D}_{i}\), \(\operatorname{Skl}^{i}S|_{\sigma}\) is homeomorphic to \(\sigma\). Also, \(\operatorname{Skl}^{1}S|_{\sigma}\) consists of only edges that connect adjacent vertices on \(\sigma\). Furthermore, property G1 holds for any output of DelPSC. This means, \(\operatorname{bd}\left(\operatorname{Skl}^{2}S|_{\sigma}\right)=\operatorname {Skl}^{1}S|_{\operatorname{bd}\sigma}\). Because of the vertex balls, we also have \(\operatorname{bd}\left(\operatorname{Skl}^{1}S|_{\sigma}\right)=\operatorname {Skl}^{0}S|_{\operatorname{bd}\sigma}\) trivially. Therefore, for \(0\leqslant i\leqslant 2\), \(\operatorname{bd}\left(\operatorname{Skl}^{i}S|_{\sigma}\right)=\operatorname {Skl}^{i-1}S|_{\operatorname{bd}\sigma}\) and \(\operatorname{Skl}^{i}S|_{\sigma}\) has vertices only in \(\sigma\).