Carathéodory balls and norm balls in a class of convex bounded Reinhardt domains
Let \(D \subset {\mathbb C}^n\) be a bounded convex domain that is balanced (i.e. \(\lambda D \subset D\) for \(\lambda \in {\mathbb C}, |\lambda|\leq 1\)). Then \(D\) defines a norm \(\mu_D\) on \({\mathbb C}^n\) such that \({\mathbb B}_{\mu_D}(0,1) = D\). Let \(c_D\) denote the Carathéodory distance on \(D\). For \(s > 0\) it is known that \({\mathbb B}_{c_D}(0,s) = {\mathbb B}_{\mu_D}(0,s^*)\) for a certain \(s^* >0\). The question which other Carathéodory ball coincides with a \(\mu_D\)-ball was first studied by \textit{B. Schwarz} in the case of \(D = \{ z\in {\mathbb C}^2: |z_1|+ |z_2|< 1\}\) [Isr. J. Math. 84, No. 1-2, 119-128 (1993; Zbl 0786.32020)]. More general cases, the so-called complex ellipsoids, were investigated by \textit{B. Schwarz} and \textit{U. Srebro} [Banach Cent. Publ. 37, 75-83 (1996; Zbl 0873.32027)], \textit{U. Srebro} [Isr. J. Math. 89, No. 1-3, 61-70 (1995; Zbl 0829.32010)], and \textit{W. Zwonek} [ Isr. J. Math. 89, No. 1-3, 71-76 (1995; Zbl 0824.32007); Ann. Pol. 64, No. 2, 183-194 (1996; Zbl 0864.32017)]. It turned out that most of the Carathéodory balls for most of the complex ellipsoids are different from \(\mu_D\)-balls. The paper under review deals with convex bounded domains of the following type \[ D = \{ z\in {\mathbb C}^n: |z_1|^{2p_1}+ 2a|z_1|^{p_1}|z_2|^{p_2} + |z_2|^{2p_2} + \sum^n_{j=3} |z_j|^{2p_j} < 1\}, \quad a> 0, p_j > 0. \] The main result is the following: a Carathéodory ball \({\mathbb B}_{c_D}(z,r)\) \((z\in D \setminus \{0\}, r>0)\) is a \(\mu_D\)-ball iff \(a=0\), exactly one of the \(p_{\ell}\)'s, say \(p_k\), is equal 1 and \(p_j = 1/2\) for \(j\not= k\), \(z_k\not= 0\) and \(z_j=0\) for \(j\not= k\). In particular, this result is the complete answer to the question from above in the case of complex ellipsoids.
scientific article; zbMATH DE number 1310630
scientific article