Extremal distance and quasiconformal reflection constants of domains in \(\mathbb{R}^ N\)
Let \(D\) be a domain in \(\overline{\mathbb{R}}^ n\) such that \(\overline{\mathbb{R}}^ n\backslash \partial D\) has exactly two components \(D\) and \(D^*\) with \(\partial D= \partial D^*\). If there is a homeomorphism \(f: \overline D\to \overline D^*\) with \(f(x)= x\) for \(x\in \partial D\), then \(\partial D\) is siad to admit a homeomorphic reflection. Set \(R_ I(D)= \inf K_ I(f)\) and \(R_ 0(D)= \inf K_ 0(f)\), where the infima are taken over all homeomorphic reflections \(f\) in \(\partial D\); \(K_ I(f)\) and \(K_ 0(f)\) stand for the inner and outer dilatation of \(f\). Another concept, which also measures how much a domain \(D\subset \overline{\mathbb{R}}^ n\) deviates from a ball or a halfspace, is the extremal distance constant \(M(D)\) of \(D\) [the author, Trans. Am. Math. Soc. 334, No. 1, 97-120 (1992; Zbl 0768.30015)]. The author studies \(M(D)\) for raylike domains, wedges, cones and cubes and \(R_ I(D)\), \(R_ 0(D)\) for some of these domains. He proves a number of general properties of domains with \(R_ I(D)< \infty\). In particular, \(\partial D\) has \(n\)-dimensional measure zero.
scientific article; zbMATH DE number 569427
scientific article