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(\mathbf{x},\mathbf{y})=\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 $O(\sqrt{\ell})$
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

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

Stefan Rass, Sandra König, Shahzad Ahmad, Maksim Goman

Submitted on 7 November 2022

## Abstract

## Preprint

Subjects: Computer Science - Computational Geometry; Computer Science - Machine Learning