La distance de Carathéodory n'est pas intérieure
Let D be a bounded domain in \({\mathbb{C}}^ n\) and \(C_ D\) denote the Carathéodory pseudodistance on D. Also, let \(C_ D^ i\) be the inner pseudodistance induced by \(C_ D\). From the definitions, it follows immediately that \(C_ D\leq C_ D^ i\) and the equality holds when D is the open unit disk in C. The distance \(C_ D\) is said to be inner if \(C_ D=C^ i_ D.\) \textit{S. Kobayashi} in his survey in Bull. Am. Math. Soc. 82, 357-416 (1976; Zbl 0346.32031), posed a question about whether \(C_ D\) is inner. T. Barth gave examples for \(C^ i_ D(p,q)>C_ D(p,q)\) showing that \(C_ D\) is not inner, and also posed a question about whether \(C_ D\) is inner if a bounded domain D is finitely compact with respect to \(C_ D\). In this article, the author constructs an example, where D is finitely compact, showing that \(C_ D\) is strictly less than \(C^ i_ D\). A result on a bounded Reinhardt domain in \({\mathbb{C}}^ 2\) is also given showing the equivalence of finite compactness, completeness and domain of holomorphy.
scientific article; zbMATH DE number 3879283