La distance de Carathéodory n'est pas intérieure
type: Item
instance of: scholarly article
title: La distance de Carathéodory n'est pas intérieure
zbMATH Open document ID: 0552.32022
DOI: 10.1007/BF03323329
published in: Results in Mathematics
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 3879283
zbMATH Keywords: Carathéodory pseudodistance
zbMATH Keywords: inner pseudodistance
zbMATH Keywords: bounded Reinhardt domain
author: Jean-Pierre Vigué
MaRDI profile type: Publication
cites work: Some Counterexamples Concerning Intrinsic Distances
cites work: Sur les fonctions de $n$ variables complexes : les transformations du produit topologique de deux domaines bornés
cites work: Q3892778
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cites work: Intrinsic distances, measures and geometric function theory
description: scientific article; zbMATH DE number 3879283