Proper holomorphic discs avoiding closed convex sets (Q1865580)

The following result is proved. Let \(\Delta\) be the open unit disc in \(\mathbb{C}\). Let \(C\) be a closed convex subset of \(\mathbb{C}^2\). Then, for any point \(p\in\mathbb{C}^2 \setminus C\) there exists a proper holomorphic map \(\varphi:\Delta \to\mathbb{C}^2\) with \(\varphi(0)=p\) and \(\varphi(\Delta)\cap C= \emptyset\) if and only if either \(C\) is a complex line or \(C\) does not contain any complex line.