A compact set with noncompact disc-hull (Q2701645)

P. Ahern and W. Rudin have studied a particular subset \(D(K)\) of the polynomially convex hull \(\widehat{K}\) of a compact subset \(K\) of \(\mathbb C^n\) The set \(D(K)\) is defined as the union of \(K\) and all \(H^{\infty}\)-discs with boundaries in \(K\). It's called the disc-hull of \(K\). In their paper [Contemp. Math. 137, 1-27 (1992; Zbl 0769.32004)], \textit{P. Ahern} and \textit{W. Rudin} asked whether \(D(K)\) is always a compact subset of \(\mathbb C^n\). NEWLINENEWLINENEWLINEThe paper under review presents a very concrete example of a connected compact set \(K\subset \mathbb C^2\), fibered over \(S^1\), such that \((0,0)\in \overline{D(K)} \setminus D(K)\). NEWLINENEWLINENEWLINERemark: Another example of a compact set \(K\) in \(\mathbb C^2\) with \(\widehat{K} = \overline{D(K)} \supsetneqq D(K)\) can be found in the paper ``A disc-hull in \(\mathbb C^2\)'' by \textit{H. Alexander} [Proc. Am. Math. Soc. 120, No. 4, 1207-1209 (1994; Zbl 0797.32010)].