Hahn, K. T.; Pflug, Peter On a minimal complex norm that extends the real Euclidean norm. (English) Zbl 0638.32005 Monatsh. Math. 105, No. 2, 107-112 (1988). The authors construct a complex norm \(N^*\) that extends the real euclidean norm and prove that it is the smallest complex norm in the following sense: If N is any complex norm in \({\mathbb{C}}^ n\) which coincides with the real euclidean norm \(| \cdot |\) in \({\mathbb{R}}^ n\) and N(z)\(\leq | z|\) for \(z\in {\mathbb{C}}^ n\), then \(N^*(z)\leq N(z)\) for \(z\in {\mathbb{C}}^ n.\) Let \(B^*_ n\) be the unit ball with respect to \(N^*\). Then \(B_ n^*\) is a convex complete circular domain with only a continuous boundary for \(n>1\). The authors obtain some complex geometric properties of \(B^*_ n:\) \(B^*_ n\) is neither biholomorphically equivalent to the unit ball \(B^ n\) nor the polydisc \(\Delta^ n\). In fact, \(B^*_ n\) is not even homogeneous. In particular, if \(n=2\), \(B^*_ 2\) is biholomorphically equivalent to \(D=\{z\in {\mathbb{C}}^ 2:| z_ 1| +| z_ 2| <1\},\) a rigid domain studied earlier by N. Kritikos [Math. Ann. 99, 321-341 (1928), JFM 54.0373.02]. Reviewer: K.T.Hahn Cited in 5 ReviewsCited in 12 Documents MSC: 32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) 32F45 Invariant metrics and pseudodistances in several complex variables 32H99 Holomorphic mappings and correspondences Keywords:minimal complex norm; convex complete circular domain; Caratheodory metric; Kobayashi metric; biholomorphically equivalent Citations:JFM 54.0373.02 PDFBibTeX XMLCite \textit{K. T. Hahn} and \textit{P. Pflug}, Monatsh. Math. 105, No. 2, 107--112 (1988; Zbl 0638.32005) Full Text: DOI EuDML References: [1] Kritikos, N.: Über analytische Abbildungen einer Klasse von vierdimensionalen Gebieten. Math. Ann.99, 321-341 (1928). · JFM 54.0373.02 [2] Rosay, J. P.: Sur une characterization de la boule parmi des domaines de ? n par son groupe d’automorphismes. Ann. Inst. Fourier, Grenoble29, 91-97 (1979). · Zbl 0402.32001 [3] Rudin, W.: Function Theory in the Unit Ball of ? n . Berlin-Heidelberg-New York: Springer. 1980. · Zbl 0495.32001 [4] Sadullaev, A.: Extremal plurisubharmonic function for the unit ball (Russian). Ann. Pol. Math.46, 433-437 (1985). · Zbl 0606.31005 [5] Siciak, J.: Extremal plurisubharmonic functions in ? n . Ann. Pol. Math.39, 175-211 (1981). · Zbl 0477.32018 [6] Vesentini, E.: Variations on a theme of Carathéodory. Ann. Scuola Norm. Sup. Pisa7, 39-68 (1979). · Zbl 0413.46039 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.