Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds

Stefan Rass, Sandra König, Shahzad Ahmad, Maksim Goman

Submitted on 7 November 2022


Given a set of points in the Euclidean space R with >1, the pairwise distances between the points are determined by their spatial location and the metric d that we endow R with. Hence, the distance d(x,y)=δ between two points is fixed by the choice of x and y and d. We study the related problem of fixing the value δ, and the points x,y, and ask if there is a topological metric d that computes the desired distance δ. We demonstrate this problem to be solvable by constructing a metric to simultaneously give desired pairwise distances between up to O() many points in R. We then introduce the notion of an ε-semimetric d~ to formulate our main result: for all ε>0, for all m1, for any choice of m points y1,,ymR, and all chosen sets of values $\{\delta_{ij}\geq 0: 1\leq i


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