Quasiconformal distortion on arcs (Q1326638)

This paper is a nicely written study of various generalizations, extensions, and ramifications of the Hayman-Gehring theorem. This theorem states that given two points \(a,b\) in a simply connected plane domain, the hyperbolic geodesic segment joining \(a\) to \(b\) is essentially the shortest among all curves joining \(a\) to \(b\). Some of the results deal with the case of domains in \(R^ n\) and in this case the quasihyperbolic metric is used in place of the hyperbolic metric.