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The Apollonian metric: limits of the comparison and bilipschitz properties. (English) Zbl 1067.30082

Author’s abstract: The Apollonian metric is a generalization of the hyperbolic metric. It is defined in arbitrary domains in \(\mathbb R^n\). In this paper, we derive optimal comparison results between this metric and the \(j_G\) metric in a large class of domains. These results allow us to prove that Euclidean bilipschitz mappings have small Apollonian bilipschitz constants in a domain \(G\) if and only if \(G\) is a ball or half-space.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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