Carathéodory balls and norm balls in \(H_ n = \{ z\in \mathbb{C}^ n:\| z\|_ 1 < 1\}\)
Let \(C\) denote the Carathéodory distance on the unit ball \(|z|_1 < 1\) for the norm \(|z|_1 = |z_1|+ \cdots + |z_n|\) on \(\mathbb{C}^n\). Since \(C(0,z) = \text{tanh}^{-1} |z|_1\), every ball for \(C\) with center 0 is also a norm ball with center 0. Moreover, when \(n = 1\), \(C\) is the Poincaré distance on the unit disk, so every ball for \(C\) is an euclidean disk, hence a norm ball, ordinarily with a different center. The author proves that these are the only situations in which an open ball for \(C\) can take the form \(|z - b|_1 < r\). This theorem was conjectured by \textit{B. Schwarz} [ibid. 84, No. 1-2, 119- 128 (1993; Zbl 0786.32020)], who proved the case \(n = 2\). Another proof has been given by \textit{W. Zwonek} [ibid. 89, No. 1-3, 71-76 (1995; Zbl 0824.32007)].
scientific article; zbMATH DE number 755537