# Adaptive Metrics for Adaptive Samples
## Appendix A Details on Metric Smoothing  
This section includes the full construction and relevant lemmas about the smoothed version of the induced metric. For a compact set \(L\subset\mathbb{R}^{d}\) and \(\beta\geq 0\), denote by \(L^{\beta}:=\{x\in\mathbb{R}^{d}\mid\min_{y\in L}\|x-y\|\leq\beta\}\) the offsets of \(L\) with respect to the Euclidean metric. The following lemma gives upper and lower bounds on the value of a smoothing of the distance-to-set function \(f_{L}\), \(\widetilde{f}_{L}\), which is defined on an arbitrarily smaller subset of Euclidean space. **Lemma A1**. : _Consider a compact set \(L\subset\mathbb{R}^{d}\). Given \(\alpha\in(0,1)\), for all \(\beta\in(0,1)\), there exists smooth function \(\widetilde{f}_{L}:\mathbb{R}^{d}\setminus L^{\beta}\to\mathbb{R}\) such that for all \(x\in\mathbb{R}^{d}\setminus L^{\beta}\), \((1-\alpha)f_{L}(x)<\widetilde{f}_{L}(x)<(1+\alpha)f_{L}(x)\)._  
Proof. : By a standard result from [18], for all \(\varepsilon>0\), there exists a smoothing \(\widetilde{f}_{L}:\mathbb{R}^{d}\setminus L^{\beta}\to\mathbb{R}\) of the distance function \(f_{L}\) such that \(\|f_{L}-\widetilde{f}_{L}\|_{\infty}<\varepsilon\). Choose \(\varepsilon=\beta\alpha\), for the given \(\alpha\in(0,1)\). By the approximation property of \(\widetilde{f}_{L}\), for all \(x\in\mathbb{R}^{d}\setminus L^{\beta}\), we have that \(f_{L}(x)-\varepsilon<\widetilde{f}_{L}(x)<f_{L}(x)+\varepsilon\). Note also that for all \(x\in\mathbb{R}^{d}\setminus L^{\beta}\), \(f_{L}(x)>\beta=\frac{\varepsilon}{\alpha}\), and thus, \(\alpha f_{L}(x)>\varepsilon\). Combining the aforementioned, we have that \(f_{L}(x)(1-\alpha)<f_{L}(x)-\varepsilon\) and \(f_{L}(x)+\varepsilon<f_{L}(x)(1+\alpha)\). Consider \(\widetilde{f}_{L}\) as defined in Lemma A1. Using this we can define a smooth adaptive distance function \(\widetilde{f}_{X}^{L}\) and provide upper and lower bounds on its value with respect to the original adaptive distance function \(f_{X}^{L}\). For \(x,y\in\mathbb{R}^{d}\setminus L^{\beta}\), we define:  
\[\widetilde{\mathrm{d}^{L}}(x,y):=\inf_{\gamma\in\mathrm{Path}(x,y)}\int_{ \gamma}\frac{dz}{\widetilde{f}_{L}(z)}\]  
and \(\widetilde{f}_{X}^{L}(y):=\widetilde{d^{L}}(y,X)\). **Lemma A2**. : _Given \(\alpha,\beta\in(0,1)\) and a smooth function \(\widetilde{f}_{L}\) defined on \(\mathbb{R}^{d}\setminus L^{\beta}\) approximating \(f_{L}\), consider a compact set \(X\subset\mathbb{R}^{d}\setminus L^{\beta}\). The Riemannian distance function \(\widetilde{f}_{X}^{\widetilde{L}}(\cdot):=\widetilde{\mathrm{d}^{L}}(\cdot,X)\) satisfies the following property for all \(y\in\mathbb{R}^{d}\setminus L^{\beta}\),_  
\[\frac{1}{1+\alpha}f_{X}^{L}(y)<\widetilde{f}_{X}^{\widetilde{L}}(y)<\frac{1}{ 1-\alpha}f_{X}^{L}(y).\]  
Proof. : Given two points \(x,y\in\mathbb{R}^{d}\setminus L^{\beta}\) and any \(\varepsilon>0\), consider \(\gamma,\gamma^{\prime}\in\mathrm{Path}(x,y)\) such that \(\mathrm{d}^{L}(x,y)\leq\int_{\gamma}\frac{dz}{f_{L}(z)}\leq\mathrm{d}^{L}(x,y )+\varepsilon\) and \(\widetilde{\mathrm{d}^{L}}(x,y)\leq\int_{\gamma^{\prime}}\frac{dz}{\widetilde {f}_{L}(z)}\leq\widetilde{\mathrm{d}^{L}}(x,y)+\varepsilon\). We then have the following inequalities resulting from inverting the inequalities in Lemma A1. \[\widetilde{\mathrm{d}^{L}}(x,y)\leq\int_{\gamma}\frac{dz}{\widetilde{f}_{L}(z )}<\frac{1}{1-\alpha}\int_{\gamma}\frac{dz}{f_{L}(z)}\leq\frac{1}{1-\alpha} \mathrm{d}^{L}(x,y)+\frac{\varepsilon}{1-\alpha},\]  
and:  
\[\frac{1}{1+\alpha}\mathrm{d}^{L}(x,y)\leq\frac{1}{1+\alpha}\int_{\gamma^{ \prime}}\frac{dz}{f_{L}(z)}<\int_{\gamma^{\prime}}\frac{dz}{\widetilde{f}_{L}( z)}\leq\widetilde{\mathrm{d}^{L}}(x,y)+\varepsilon.\]  
Since these equalities hold for all \(\varepsilon>0\), then we can conclude that for all pairs \(x,y\in\mathbb{R}^{d}\setminus L^{\beta}\), \(\frac{1}{1+\alpha}\mathrm{d}^{L}(x,y)<\widetilde{\mathrm{d}^{L}}(x,y)<\frac{1 }{1-\alpha}\mathrm{d}^{L}(x,y)\). Now, consider \(y\in\mathbb{R}^{d}\setminus L^{\beta}\).