Peak points in boundaries not of finite type (Q1057400)

It is known that, in domains in \({\mathbb{C}}^ 2\) which are pseudoconvex and of finite type, compact subsets of peak sets for \(A^{\infty}(D)\) are peak sets for \(A^{\infty}(D)\). We give an example of a convex domain D (not of finite type) whose weakly pseudoconvex boundary points form a line segment K with the property: K is a peak set for \(A^{\infty}(D)\), but a point \(p\in K\) is not a peak point for any \(A^{\alpha}(D)\), \(\alpha >0\). We also consider briefly the case when the weakly pseudoconvex boundary points form a disc.