Hyperbolic distances in Hilbert spaces

We present a functional equations approach to the non-negative functions h (x,y) and E (x,y) satisfying cosh h (x, y) = √ 1 + x2 √ 1+ y2 -xy, E (x,y) = ||x -y||. The underlying structure is a pre-Hilbert space X of dimension at least 2. An important tool is the group of translations Tt(x) = x + ((xe)(cosh t - 1) + √1 + x2 sinh t) e, t ε ℝ, where Tt : X → X satisfies the translation equation with a fixed e ∈ X such that e2 = 1. One of the results is that a function d : X × X → ℝ≥0 := {r ε ℝ | r ≥ 0} which is invariant under orthogonal mappings and the described translations for a fixed e, must be of the form d (x,y) = g ((h(x,y)} with an arbitrary function g : ℝ≥0 → ℝ≥0- If, moreover, d is additive on the line (ℰe | ℰ ∈ ℝ}, then d is essentially equal to h. © Birkhäuser Verlag, Basel, 1999.