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Polynomial hulls with convex sections and interpolating spaces. (English) Zbl 0588.32017
Für ein Kompaktum \(L\subset \partial D\times\mathbb C^ m\), für das alle Fasern \(L(z)=\{w\in\mathbb C^ m: (z,w)\in L\},\) \(z\in \partial D\), nicht leere konvexe Kompakta sind, wird die polynomkonvexe Hülle \(K\) beschrieben; \(D\) bezeichne hier den Einheitskreis der komplexen Ebene. Enthält \(K\) Punkte über \(D\), so wird gezeigt, daß \(K\setminus (\partial D\times\mathbb C^ m)\) durch Graphen beschränkter analytischer Abbildungen \(f: D\to\mathbb C^ m\) ausgeschöpft werden kann. Ohne die Voraussetzung an die Fasern enthält \(K\setminus (\partial D\times\mathbb C^ m)\) i.A. keine analytischen Scheiben [vgl. J. Wermer, Ark. Mat. 20, 129–135 (1982; Zbl 0491.32013)].
Der hier gegebene Beweis beruht auf der Interpolationstheorie für Familien endlich-dimensionaler normierter Räume [vgl. R. Coifman, R. Rochberg, G. Weiss, M. Cwikel und Y. Sagher, Lect. Notes Math. 779, 123–153 (1980; Zbl 0427.46049)]. Ein anderer Beweis für \(m=1\) findet sich in der Arbeit von H. Alexander und J. Wermer [Math. Ann. 271, 99–109 (1985; Zbl 0538.32011)].

MSC:
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
46M35 Abstract interpolation of topological vector spaces
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[1] Herbert Alexander and John Wermer, On the approximation of singularity sets by analytic varieties, Pacific J. Math. 104 (1983), no. 2, 263 – 268. · Zbl 0543.32005
[2] Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99 – 109. · Zbl 0538.32011 · doi:10.1007/BF01455798 · doi.org
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