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Let H denote the standard one-point completion of a real Hilbert space. Givenany 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 showthat their associated Apollonian metrics satisfy the following uniformcontraction principle: Let D=diam_V (U) \in [0,+\infty] be the diameter of thesmaller subsets with respect to the large. Then for every x,y in U we haved_V(x,y) \leq tanh (D/4) d_U(x,y).  In dimension one, this contraction principle was established by Birkhoff forthe Hilbert metric of finite segments on RP^1. In dimension two it was shown byDubois for subsets of the Riemann sphere. It is new in the generality statedhere.

A uniform contraction principle for bounded Apollonian embeddings