ANNALES POLONICI MATHEMATICI 82.3 (2003) Some properties of Reinhardt domains by Le Mau Hai, Nguyen Quang Dieu and Nguyen Huu Tuyen (Hanoi) Abstract. We first establish the equivalence between hyperconvexity of a fat bounded Reinhardt domain and the existence of a Stein neighbourhood basis of its closure. Next, we give a necessary and sufficient condition on a bounded Reinhardt domain D so that every holomorphic mapping from the punctured disk Δ * into D can be extended holomorphically to a map from Δ into D. 1. Introduction. Let D be a domain in C n . We say that D is a Rein- hardt domain if D is invariant under the action of the n-torus (for a precise definition see Section 2). Reinhardt domains are important objects in com- plex analysis; their pseudoconvexity, hyperconvexity, kinds of hyperbolicity, etc. have been characterized in [CCW], [Zw1], [Zw2], etc. The aim of this paper is to study Reinhardt domains in connection with other concepts. Namely, in Theorem 3.2 we prove that a fat Reinhardt do- main is hyperconvex if and only if its closure is compact Stein, i.e. has a neighbourhood basis of Stein domains. It should be remarked that there ex- ists a fat, pseudoconvex domain in C 2 whose closure is polynomially convex but the domain itself is not hyperconvex (see Proposition 3.1). On the other hand, the “worm” domains constructed by Diederich and Fornæss provide examples of hyperconvex domains whose closure is not compact Stein. In Section 4, we deal with the question of extending holomorphic map- ping into Reinhardt domains. Roughly speaking, we say that a domain D in C n has the k- or ∆ ∗ -extension property if every holomorphic mapping into D can be holomorphically extended through a “small” set (see Sec- tion 4 for precise definitions). The ∆ ∗ -extension property was first studied by D. D. Thai [T] and recently by P. Thomas and D. D. Thai [TT]. In general, the k-extension property is strictly stronger than the ∆ ∗ -extension property. 2000 Mathematics Subject Classification : 32A07, 32D15, 32T05. Key words and phrases : Reinhardt domain, pseudoconvex domain, hyperconvex do- main, plurisubharmonic function. [203]