Metric space inversions, quasihyperbolic distance, and uniform spaces

Abstract

We define a notion of inversion valid in the general metric space setting.We establish several basic facts concerning inversions; e.g., they are quasimöbius homeomorphisms and quasihyperbolically bilipschitz.In a certain sense, inversion is dual to sphericalization.We demonstrate that both inversion and sphericalization preserve local quasiconvexity and annular quasiconvexity as well as uniformity.