The average scale-invariant Cassinian metric is sharply comparable to four standard hyperbolic-type metrics, and its balls in punctured Euclidean space are convex exactly for radius at most log 3.
Visual angle metric in the upper half plane
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abstract
We prove an identity which connects the visual angle metric $v_{\mathbb{H}^2}$ and the hyperbolic metric $\rho_{\mathbb{H}^2}$ of the upper half plane $\mathbb{H}^2$. The proof is based on geometric arguments and uses computer algebra methods for formula manipulation. We also prove a sharp H\"older continuity result for quasiregular mappings with respect to the visual angle metric.
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On the average scale-invariant Cassinian metric
The average scale-invariant Cassinian metric is sharply comparable to four standard hyperbolic-type metrics, and its balls in punctured Euclidean space are convex exactly for radius at most log 3.