Remarks on Lempert functions of balanced domains (Q1027748)

For \(D \subset \mathbb C^n\) a balanced domain (\(z \in D\), \(\lambda \in \mathbb C\), \(|\lambda| \leq 1\) implies \(\lambda z \in D\)), let \(c_D\) denote the Carathéodory pseudodistance (defined as a supremeum over holomorphic maps \(f : D \to \mathbb D\), \(\mathbb D\) the unit disc), \(k_D\) the Kobayashi pseudodistance, \(h_D(z) = \inf \{ t > 0 : z/t \in D\}\) the Minkowski function of \(D\), \(\hat{D}\) the convex hull of \(D\) and \(\hat{h}_D = h_{\hat D}\). Recall that \(k_D\) is defined in terms of holomorphic \(\phi : \mathbb D \to D\), but one must modify the function \(k_D^{(1)}(z, w) = \inf \{ \tanh^{-1} |\phi(\alpha)| : \phi(0) = z, \phi(\alpha) = w\}\) to produce a quantity that satisfies the triangle inequality. The \(m\)-th Lempert function \(k_D^{(m)}\) is defined inductively via \(k_D^{(m+1)}(z, w) = \inf \{k_D^{(m)}(z, z') + k_D^{(1)}(z', w) : z' \in D\}\), and \(k_D = \inf_m k_D^{(m)}\). It is known that \(\tanh^{-1} \hat{h}_D(z) \leq c_D(0, z) \leq k_D(0, z) \leq k_D^{(1)}(0, z) \leq \tanh^{-1} h_D(z)\) holds for all \(z \in D\), and that \(a \in D\) satisfies \(k_D(0, a) =\tanh^{-1} h_D(a)\) if and only if \(\hat{h}_D(a) = h_D(a)\). The first new result here is that this is also equivalent to \(k_D^{(3)}(0, a) = \tanh^{-1} h_D(a) \). While there is an example given where this fails for \(k_D^{(1)}\) in place of \(k_D^{(3)}\), it is not resolved whether it could hold in general with \(k_D^{(2)}\). Some of the other results compare \(D\) with pseudoconvex balanced \(G \supset D\) or impose the assumption that \(D\) is taut. Several intriguing examples are also given, including a non-convex pseudoconvex balanced domain \(D \subset \mathbb C^2\) with \(\hat{D} = \mathbb B_2\) (the unit ball) such that \(k_D^{(2)}(0, z) = \tanh^{-1} \|z\|\) for \(z \in D\).