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# Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds

Given a set of points in the Euclidean space ${\mathbb{R}}^{\ell }$ with $\ell >1$, the pairwise distances between the points are determined by their spatial location and the metric $d$ that we endow ${\mathbb{R}}^{\ell }$ with. Hence, the distance $d\left(\mathbf{x},\mathbf{y}\right)=\delta$ between two points is fixed by the choice of $\mathbf{x}$ and $\mathbf{y}$ and $d$. We study the related problem of fixing the value $\delta$, and the points $\mathbf{x},\mathbf{y}$, and ask if there is a topological metric $d$ that computes the desired distance $\delta$. We demonstrate this problem to be solvable by constructing a metric to simultaneously give desired pairwise distances between up to $O\left(\sqrt{\ell }\right)$ many points in ${\mathbb{R}}^{\ell }$. We then introduce the notion of an $\epsilon$-semimetric $\stackrel{~}{d}$ to formulate our main result: for all $\epsilon >0$, for all $m\ge 1$, for any choice of $m$ points ${\mathbf{y}}_{1},\dots ,{\mathbf{y}}_{m}\in {\mathbb{R}}^{\ell }$, and all chosen sets of values \$\{\delta_{ij}\geq 0: 1\leq i