Abstract
Let D be a non-pseudoconvex open set in \({\mathbb{C}^n}\) and S be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with D. Sufficient conditions are given for \({\mathbb{C}^3{\setminus} S}\) to belong to a complex line. Moreover, in the \({{\mathcal{C}}^2}\)-smooth case, it is shown that \({S=\mathbb{C}^n}\).
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This paper was written during the stay of the first-named author at the Carl von Ossietzky Universität, Oldenburg (February–March 2010) supported by a DFG grant 436Pol113/106/2. The authors would like to thank Pascal J. Thomas for helpful discussions about Proposition 1 and for pointing out a serious mistake in the proof of Lemma 3 in a former version of this paper.
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Nikolov, N., Pflug, P. Two-dimensional slices of non-pseudoconvex open sets. Math. Z. 272, 381–388 (2012). https://doi.org/10.1007/s00209-011-0938-z
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DOI: https://doi.org/10.1007/s00209-011-0938-z