Convex domains of finite type (Q1803319)

Let \(\Omega\subset\subset\mathbb{C}^ n\) be a smoothly bounded domain and \(p\in\partial\Omega\). Let \(p\) have a neighbourhood \(U\) in which \(\Omega\) is convex. Suppose that the line type of \(p\) is \(L<\infty\). The author proves that for each \(z\in\Omega\cap U\), there exists a uniformly bounded \(C^ \infty\)-plurisubharmonic function on \(\Omega\) with maximally large Hessian on a polydisc \(P_ \delta(z)\). As a consequence, it is deduced that the variety type of \(p\) is also finite and equals \(L\). This corollary is also known to Fornaess-Sibony and Boas-Straube by different methods.