## 7 Proofs of lemmas in Section 2
Without loss of generality, assume that \(A\) has end points of both \(C\) and \(C^{\prime}\) and let \(s\) and \(t\) be these end points. We observe that the line segment \(st\) cannot make an angle with \(L\) less than the least angle made between \(L\) and a tangent to \(A\). This least angle is made by the tangent to \(A\) at any of its end points. Since the width of \(L^{*}\) is \(0.07\lambda\) and the radius of \(H_{F}\cap B\) is at least \(\sqrt{3}\lambda\), this least angle is more than \(\frac{\pi}{2}-\arcsin(0.035/\sqrt{3})\). Therefore, \(st\) makes an angle of at least \(\frac{\pi}{2}-\arcsin(0.035/\sqrt{3})\) with \(L\), or equivalently of at most \(\arcsin(0.035/\sqrt{3})<0.03\) with \(\vec{d}\). Since\(\angle\vec{d},\mathbf{n}_{s}\leqslant\angle\vec{d},\mathbf{n}_{p}+\angle\mathbf{n}_{p },\mathbf{n}_{s}\leqslant\arcsin(1/16)+(1/8)\leqslant 0.2\), \(st\) makes an angle of at most \(0.23\) radians with \(\mathbf{n}_{s}\). The line of \(st\) intersects \(\sigma\) at two points, namely at \(s\) and \(t\), where \(\angle st,\mathbf{n}_{s}\leqslant 0.23\). Here we can use the Long Distance Lemma in [21] to obtain \(d(s,t)\geqslant 2\rho_{\sigma}\cos 0.23\geqslant 62\lambda\) which contradicts \(d(s,t)\leqslant 2\times 0.07\lambda=0.14\lambda\). Next, consider the case where \(C\) has both endpoints on a single arc, say \(A\). Taking these end points as \(s\) and \(t\) we can argue as in the previous paragraph to reach a contradiction. Third, we claim that for any facet \(F\) of \(V_{p}\), if \(F\) intersects \(B\cap\sigma\), \(F\cap B\cap\sigma\) is a single open curve with endpoints in \(\operatorname{bd}F\) or in \(\operatorname{bd}\sigma\). We already proved that there is no closed curve in \(F\cap B\cap\sigma\). Since \(F\) does not have any tangential contact with \(\sigma\), \(F\cap B\cap\sigma\) is a set of open curves and the endpoints of any open curve in \(F\cap\sigma\) thus lie in \(\operatorname{bd}F\) or in \(\operatorname{bd}\sigma\). Assume to the contrary that \(F\cap B\cap\sigma\) contains two open curves, say \(\xi\) and \(\xi^{\prime}\). By our assumption, \(H_{F}\) is within a distance of \(\lambda\) from \(p\). We have shown before that \(H_{F}\cap B\cap\sigma\) is a single open curve. Follow \(H_{F}\cap B\cap\sigma\) from \(\xi\) to \(\xi^{\prime}\). When we leave \(\xi\), we must leave \(F\) at a Voronoi edge \(e\subset\operatorname{bd}F\). Afterwards, we stay in the plane \(H_{F}\) and we must cross the support line of \(e\) again in order to reach \(\xi^{\prime}\). Therefore, some tangent to \(H_{F}\cap B\cap\sigma\) is parallel to \(e\). However, the angle between the surface normal at some point on \(H_{F}\cap B\cap\sigma\) and \(\mathbf{n}_{p}\) would then be at least \(\pi/2-\angle\mathbf{n}_{p},e\geqslant\pi/2-5\sqrt{2(\lambda/\rho_{\sigma})^{2}}\) by Lemma 10(ii) since \(\varepsilon,\omega\leqslant\lambda/\rho_{\sigma}\). This contradicts Lemma 10(i.a) proving the claim. Recall that \(\sigma_{p}\) denote the components of \(V_{p}|_{\sigma}\) that intersect a Voronoi edge of \(V_{p}\). We show that \(\sigma_{p}\) intersect \(V_{p}\) and its faces in topological balls if \(S\) has the \(\lambda\)-size property for a sufficiently small \(\lambda\). We need the following result which says that when \(\lambda\) is sufficiently small, each Voronoi cell \(V_{p}\) contains at least one Voronoi edge intersecting \(\sigma\) if \(p\in\sigma\in\mathcal{D}_{2}\). **Lemma 13**: _There exists a \(\lambda>0\) so that if \(S\) satisfies the \(\lambda\)-size property, then \(\sigma_{p}\) is non-empty for any \(p\in\sigma\)_  
Suppose that \(\sigma_{p}\) is empty, that is, no Voronoi edge of \(V_{p}\) intersects \(\sigma\). By our assumption \(\operatorname{bd}\sigma\) is non-empty and let \(q\in\sigma\) be a weighted point in \(\operatorname{bd}\sigma\). Because of Lemma 1, an edge of \(V_{q}\) has to intersect \(\sigma\). Consider walking on a path in \(\sigma\) from \(p\) to \(q\). Let \(p=p_{0},p_{1},..,p_{k}=q\) be sequence of vertices whose Voronoi cells are encountered along this walk.