A uniform contraction principle for bounded Apollonian embeddings

TL;DR

This paper establishes a new uniform contraction principle for Apollonian metrics in nested subsets of a Hilbert space, generalizing previous results in one and two dimensions to a broader setting.

Contribution

It introduces a general uniform contraction principle for Apollonian metrics in Hilbert spaces, extending known results from lower dimensions to infinite-dimensional spaces.

Findings

The Apollonian metrics satisfy a contraction bound involving the hyperbolic tangent function.

The result generalizes Birkhoff's and Dubois' contraction principles to higher dimensions and more general subsets.

The contraction depends on the diameter of the smaller set within the larger subset.

Abstract

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.