A uniform contraction principle for bounded Apollonian embeddings

Let H denote the standard one-point completion of a real Hilbert space. Given any non-trivial proper sub-set U of H one may define the so-called `Apollonian' metric d_U on U. When U \subset V \subset H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let D=diam_V (U) \in [0,+\infty] be the diameter of the smaller subsets with respect to the large. Then for every x,y in U we have d_V(x,y) \leq tanh (D/4) d_U(x,y). In dimension one, this contraction principle was established by Birkhoff for the Hilbert metric of finite segments on RP^1. In dimension two it was shown by Dubois for subsets of the Riemann sphere. It is new in the generality stated here.