Hyperbolic and euclidean distance functions

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).