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Quasihyperbolic metrics, $$\text{Lip}_G$$-extension domains and John disks. (English) Zbl 1254.30070
Summary: Suppose that $$D$$ is a Jordan proper subdomain of $$\mathbb R^2$$. In this paper, we prove that (1) $$D$$ is a John disk if and only if there exists a constant $$c \geq 1$$ such that
$k_D(x1, x2) \leq cG_D(x1, x2)\qquad\text{for all}\qquad x_1, x_2 \in D;$ (2) $$D$$ is a John disk if and only if $$D$$ is a $$\text{Lip}_G$$-extension domain. Here $$k_D$$ is the quasihyperbolic metric in $$D$$,
$G_D(x_1, x_2) = \frac{1}{2} \log\bigg(1 + \frac{l(\gamma )}{ d(x_1,\partial D)} \bigg)\bigg(1 + \frac{l(\gamma )}{ d(x_2,\partial D)}\bigg),$
$$\gamma \subset D$$ is the quasihyperbolic geodesic with endpoints $$x_1$$ and $$x_2$$, and $$l(\gamma )$$ is the Euclidean length of $$\gamma$$.
##### MSC:
 30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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