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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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