Non-linearity of the pluricomplex Green function (Q2701581)

By a deep theorem due to Lempert, the pluricomplex Green function \(g(z,w)\) of a convex domain \(\Omega\subset\mathbb{C}^n\) with pole at \(w\in \Omega\) coincides with the function \(\delta(z,w)= \inf\log|f^{-1}(w)|\), the infimum being taken over all holomorphic mappings \(f\) from the unit disc \(\Delta \subset \mathbb{C}\) to \(\Omega\) such that \(f(0)=z\) and \(w\in f(\Delta)\). To deal with the Green functions with several poles, the author extends the notion of the Lempert function \(\delta\). Namely, let \(A=\{(w_j, \nu_j)\}_j\) be a finite subset of \(\Omega\times \mathbb{R}_+\), then \(\delta(z,A): =\inf\sum_j \nu_j \log |f^{-1} (w_j)|\), where \(f(0)=z\) and \(w_j\in f(\Delta)\). Properties of \(\delta(z,A)\) and its relations to the Green function \(g(z,A)\) are studied. In particular, the set \(\{z:\delta (z,A)=\sum_j \nu_j\delta (z,w_j)\}\) is described as the union of all complex geodesics (for the Kobayashi metric) passing through all the points \(w_j\).