A Uniform Contraction Principle for Bounded Apollonian Embeddings

Let ̂ H = H ∪ {∞} denote the standard one-point completion of a real Hilbert space H. Given any non-trivial proper subset U ⊂ ̂ H one may define the so-called “Apollonian” metric dU on U . When U ⊂ V ⊂ ̂ H are nested proper subsets we show that their associated Apollonian metrics satisfy the following uniform contraction principle: Let Δ = diamV (U) ∈ [0,+∞] be the diameter of the smaller subsets with respect to the large. Then for every x, y ∈ U we have dV (x, y) ≤ tanh Δ 4 dU (x, y). In dimension one, this contraction principle was established by Birkhoff [Bir57] for the Hilbert metric of finite segments on RP. In dimension two it was shown by Dubois in [Dub09] for subsets of the Riemann sphere ̂ C ∼ ̂ R2. It is new in the generality stated here.