Old and New on the Quasihyperbolic Metric

Let D be a proper subdomain of $$ {\mathbb{R}^d}$$ . Following Gehring and Palka [GP] we define the quasihyperbolic distance between a pair x 1, x 2 of points in D as the infimum of $$ {\smallint _\gamma }\frac{{ds}}{{D\left( {x,\partial D} \right)}}$$ over all rectifiable curves γ joining x 1, x 2 in D. We denote the quasihyperbolic distance between x 1, x 2 by k D (x 1, x 2). As pointed out by Gehring and Osgood [GO], x 1 and x 2 can be joined by a quasihyperbolic geodesic; also see [Mr]. The quasihyperbolic metric is comparable to the usual hyperbolic metric in a simply connected plane domain by the Koebe distortion theorem. For a multiply connected plane domain D these two metrics are comparable if and only if the boundary of D is uniformly perfect as shown by Beardon and Pommerenke [BP]. Gehring and Palka introduced the quasihyperbolic metric as a tool for the study of quasiconformal homogeneity and since then this metric has found a number of applications. Let us simply mention here that this metric plays an important role in the famous extension theorem of Jones [J1] for functions in BMO and that this metric has by now become a standard tool in the study of quasiconformal mappings.