What's so special about Euclidean distance?

In this paper, we provide an application-oriented characterization of a class of distance functions monotonically related to the Euclidean distance in terms of some general properties of distance functions between real-valued vectors. Our analysis hinges upon two fundamental properties of distance functions that we call "value-sensitivity" and "order-sensitivity". We show how these two general properties, combined with natural monotonicity considerations, lead to characterization results that single out several versions of Euclidean distance from the wide class of separable distance functions. We then discuss and motivate our results in two different and apparently unrelated application areas-mobility measurement and spatial voting theory-and propose our characterization as a test for deciding whether Euclidean distance (or some suitable variant) should be used in your favourite application context.