Hyperbolic and euclidean distance functions

Abstract

This is a functional equations approach to the non-negative functions $h(x,y)$ and $e(x,y)$ as defined in formulas (1) and (2). Moreover, all distance functions of ${\mathbb{R}}_n$ are characterized, which are invariant under linear and orthogonal mappings (see Theorem 1), and, especially, all functions of this type are determined, which satisfy in addition ($D_2$) (see Theorem 2). Here ($D_2$) asks for the invariance under euclidean or hyperbolic translations of the $x_1$-axis. Finally, additivity on the $x_1$-axis is considered, leading to the distance functions $h$ and $e$ up to non-negative factors (see Theorem 3).