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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… (More)

We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.

We characterize the class of uniform domains in terms of capacity. As a byproduct of this investigation we provide results describing when a Loewner domain will be QED.

We characterize uniform spaces in terms of a slice condition. We also establish the Gehring-Osgood-Väisälä theorem for uniform spaces in the metric space context.

- David A. Herron
- The American Mathematical Monthly
- 2012

We use the intrinsic diameter distance to describe when a Riemann map has a continuous extension to the closed unit disk. I

We document various properties of the classes of locally uniform and weakly linearly locally connected domains. We describe the boundary behavior for quasiconformal ho-meomorphisms of these domains and exhibit certain metric conditions satissed by such maps. We characterize the quasiconformal homeomorphisms from locally uniform domains onto uniform domains.… (More)

- DAVID A. HERRON
- 2016

We characterize the open sets in the sphere that are geodesically convex in any containing domain with respect to various conformal metrics.

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