The Apollonian inner metric
Let \(G\) be a domain in \(\mathbb R^n\), where \(G\) is a proper subset of \(\mathbb R^n\). For \(x,y \in G\), the Apollonian metric is defined by \(\alpha_G(x,y)=\sup_{a,b\in \partial G}\log\frac{| a-x |}{| a-y |}\frac{| b-y |}{| b-x |}\). This metric is known for analysts as the Apollonian metric, introduced and first studied by \textit{A. F. Beardon} [in: Quasiconformal mappings and analysis, New York, Springer, 91--108 (1998; Zbl 0890.30030)], and for geometers as the Barbilian metric. To any metric \(d\) on \(G\) one may associate an {inner metric} \(\tilde{d}\), with \(\tilde{d}(x,y)\) being defined as the infimum of lengths of curves joining \(x\) and \(y\) inside \(G\). Some highlights of this study of the inner metric \(\tilde{\alpha}_G\) associated with the Apollonian metric \(\alpha_G\): (1) \(\tilde{\alpha}_G\) is a pseudometric for every proper domain of \(\mathbb R^n\), and a metric if and only if the complement \(G^c\) of \(G\) is not contained in an \((n-2)\)-dimensional plane; (2) if \(G^c\) is not contained in a hyperplane, then there exists a geodesic connecting any two points in \(G\).
scientific article; zbMATH DE number 2148048