Euclidean Distance Between Intervals Is the Only Representation-Invariant One

Authors

Olga Kosheleva, University of Texas at El PasoFollow
Vladik Kreinovich, University of Texas at El PasoFollow

Publication Date

7-2020

Comments

Technical Report: UTEP-CS-20-83b

Published in Reliable Computing, 2021, Vol. 28, pp. 4-9.

Abstract

An interval can be represented as a point in a plane, e.g., as a point with its endpoints as coordinates. We can thus define distance between intervals as the Euclidean distance between the corresponding points. Alternatively, we can describe an interval by its center and radius, which leads to a different definition of distance. Interestingly, these two definitions lead, in effect, to the same distance -- to be more precise, these two distances differ by a multiplicative constant. In principle, we can have more general distances on the plane. In this paper, we show that only for Euclidean distance, the two representations lead to the same distance between intervals.