Continuity of conformal capacity for~condensers with uniformly perfect plates (Q1963377)

A compact set \(A\) in \(\overline{\mathbb R}^n\) is \(\alpha\)-uniformly perfect if the moduli of the ring domains separating \(A\) are bounded from above by \(\alpha\). Denote by \(UP(\alpha)\) the class of all \(\alpha\)-uniformly perfect compact subsets of \(\overline{\mathbb R}^n\). Basic properties of \(UP(\alpha)\) were studied by \textit{P.~Järvi} and \textit{M.~Vuorinen } [J. Lond. Math. Soc., II. Ser. 54, No. 3, 515-529 (1996; Zbl 0872.30014)]. In the article under review, the classical Väisäla lemma is proven with continua replaced by uniformly perfect sets. The author defines the \(\alpha\)-inner diameter of sets with respect to the class \(UP(\alpha)\) and introduces the notion of strong convergence of sets. A theorem on continuity of the conformal capacity is established on the base of these notions. In addition, it is proven that a homeomorphism \(f\:\overline{\mathbb R}^n \to \overline{\mathbb R}^n\) is quasiconformal if and only if both \(f\) and \(f^{-1}\) preserve the class \(UP=\bigcup_{\alpha>0}UP(\alpha)\).