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Convex domains of finite type. (English) Zbl 0777.31007
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.
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