PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 5, Pages 1473–1475 S 0002-9939(00)05704-X Article electronically published on October 25, 2000 A COMPACT SET WITH NONCOMPACT DISC-HULL BUMA FRIDMAN, LOP-HING HO, AND DAOWEI MA (Communicated by Steven R. Bell) Abstract. The disc-hull of a set is the union of the set and all H ∞ discs whose boundaries lie in the set. We give an example of a compact set in C 2 whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin. The polynomial hull of a compact set X ⊂ C n is the set  X of all points x ∈ C n at which the inequality |P (x)|≤ max{|P (z )| : z ∈ X } holds for every polynomial P . Let U denote the unit disc in C. In [1] P. Ahern and W. Rudin introduced the following definition. “If Φ : U → C n is a non-constant map whose components are in H ∞ (U ), its range Φ(U ) is called an H ∞ -disc, parametrized by Φ. If lim rր1 (Φ(re iθ )) ∈ X for almost all e iθ on the unit circle T , then Φ(U ) is an H ∞ -disc whose boundary lies in X.” They further define the disc-hull D(X ) to be the union of X and all H ∞ -discs whose boundaries lie in X. Because of the maximum principle, D(X ) ⊂  X. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X ) is always compact for a compact set X ⊂ C n . Below we answer this question negatively by constructing a counter-example in C 2 . 1. Define ω = {z ∈ U : Re z> 1 2 }. Let ϕ : U → ω be the Riemann map satisfying ϕ(±i)= 1 2 ± √ 3 2 i and ϕ(1) = 1. Therefore Re ϕ(e iθ )= 1 2 for Re e iθ ≤ 0. Also, 0 / ∈ ϕ( U ), | ϕ(0) |< 1, and hence lim n→∞ ϕ n (0) = 0. 2. Let X = {(ζ,η) ∈ C 2 : ζ ∈ T,η ∈ Γ ζ }, where the fiber Γ ζ is defined as follows. Γ ζ = T for Re ζ> 0, Γ ζ = U for ζ = ±i, and Γ ζ = {ϕ n (ζ ): n ∈ N}∪{0} for Re ζ< 0. One can check that the complement C 2 \X of X is an open set and, since X is also bounded, it is compact. One can also notice that X is connected. 3. For each n consider Φ n (z )=(z,ϕ n (z )) : U → C 2 . By construction, Φ n (T ) ⊂ X , so, Φ n (U ) ∈ D(X ). One can see that lim n→∞ Φ n (0) = (0, 0); therefore (0, 0) ∈ D(X ). Received by the editors August 31, 1999. 2000 Mathematics Subject Classification. Primary 32E20. Key words and phrases. Polynomial convexity, disc-hull. c 2000 American Mathematical Society 1473 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use