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The polynomial hull of a rectifiable curve in $${\mathbb{C}}^ n$$. (English) Zbl 0659.32017
For a compact connected set X in $${\mathbb{C}}^ n$$, denote by $$\hat X$$ the polynomially convex hull of X. The author proves the following
Theorem: Let X be a rectifiable Jordan curve in $${\mathbb{C}}^ n$$ with $$V=\hat X\setminus X$$ nonempty. Then V is an irreducible analytic subset (of pure dimension one) of $${\mathbb{C}}^ n\setminus X$$ of finite area. More precisely $4\pi area(V)\leq L^ 2,$ where L is the length of X.
When X is a smooth Jordan curve, the finiteness of the area of V is a special case of general results of N. Sibony [Duke Math. J. 52, 157-197 (1985; Zbl 0578.32023)] on positive currents. If X is only assumed to be a connected compact set of finite linear measure, it is not known whether the hull has finite area.
Reviewer: E.Straube

##### MSC:
 32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables 32C25 Analytic subsets and submanifolds 32C30 Integration on analytic sets and spaces, currents
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