A note on \(\phi\)-uniform domains
Let \(\Omega\) be a domain in \(\mathbb R^n\) and \(d(x)\) be the euclidean distance of a point in \(\Omega\) to its boundary. A domain \(\Omega\) is uniform iff the quasihyperbolic distance \(k(x,y)\) satisfies the inequality \(k(x,y) \leq C \log \left(1+\frac{|x-y|}{d(x)\wedge d(y)} \right)\) for all \(x,y \in \Omega\) (\(C\geq 1\)). Let \(\phi: [0,\infty) \to [0,\infty)\) be a homeomorphism. A domain \(\Omega\) is said to be \(\phi\)-uniform if \(k(x,y) \leq \phi \left( \frac{|x-y|}{d(x)\wedge d(y)} \right)\) for all \(x,y \in \Omega\). The author constructs an example of a quasiconvex, simply connected and bounded planar domain which fails to be \(\phi\)-uniform for any homeomorphism.
scientific article; zbMATH DE number 6710824