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The length of a set in the sphere whose polynomial hull contains the origin. (English) Zbl 0762.32007
Definition. (a) A set \(X<\mathbb{R}^ k\) is 1-rectifiable if it is the image of a bounded subset \(U\subset\mathbb{R}\) under a Lipschitz continuous mapping \(f:U\to\mathbb{R}^ k\).
(b) \(X\) is \(({\mathcal H}^ 1,1)\)-rectifiable if \({\mathcal H}^ 1(X)<\infty\) and \({\mathcal H}^ 1\)-almost all of \(X\) can be covered by a countable union of 1-rectifiable sets.
\({\mathcal H}^ 1\) denotes here a 1-dimensional Hausdorff measure.
The main result of the paper is Theorem. If \(X\) is a compact \(({\mathcal H}^ 1,1)\)-rectifiable subset of the unit sphere \(S\subset\mathbb{C}^ n\) such that the origin \(0\in\mathbb{C}^ n\) belongs to the polynomial hull \(\widehat X\), then \({\mathcal H}^ 1(X)\geq 2\pi\).

MSC:
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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[1] Alexander, H., Polynomial approximation and hulls in sets of finite linear measure in \(C\)^{n}, Amer. J. math., 93, 65-74, (1971) · Zbl 0221.32011
[2] Alexander, H., The polynomial hull of a set of finite linear measure in \(C\)^{n}, J. analyse math., 47, 238-242, (1986) · Zbl 0615.32009
[3] Alexander, H., The polynomial hull of a rectifiable curve in \(C\)^{n}, Amer. J. math., 110, 629-640, (1988) · Zbl 0659.32017
[4] Bishop, E., Conditions for analyticity, Michigan math. J., 11, 289-304, (1964) · Zbl 0143.30302
[5] Burago, Yu.D.; Zalgaller, V.A., Geometric inequalities, Grundlehren der math. wiss., 285, (1988), Springer Berlin-Heidelberg-New York · Zbl 0633.53002
[6] Chirka, E.M., Complex analytic sets. math. and its appl., 46, (1989), Kluwer Dordrecht, Original in Russian, Nauka, Moskva (1985)
[7] Federer, H., Geometric measure theory, Grundlehren der math. wiss., 153, (1969), Springer Berlin-Heidelberg-New York · Zbl 0176.00801
[8] Fornaess, J.E., The several complex variables problem List, Ann arbor, 10, (1991), Sept.
[9] Sibony, N., Valeurs au bord de fonctions holomorphes et ensembles polynomialement convexes, Lecture notes in math, 578, (1977), Springer Berlin-Heidelberg-New York · Zbl 0382.32004
[10] Stolzenberg, G., Polynomially and rationally convex sets, Acta math., 109, 259-289, (1963) · Zbl 0122.08404
[11] Stout, E.L., Removable sets for holomorphic functions of several complex variables, Publ. mat., 33, 345-362, (1989) · Zbl 0692.32010
[12] Alexander, H., Quasi-isoperimetric inequality for polynomial hulls, (1992), Preprint · Zbl 0794.32017
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