Rigid characterizations of pseudoconvex domains (Q2841857)

For an open set \(D\subset\mathbb C^n\) and a point \(z\in D\) let \(I_{D,z}\) denote the maximal balanced domain such that \(B_{D,z}:=z+I_{D,z}\subset D\). Put \(H_D:=\{(z,w)\in D\times\mathbb C^n: w\in I_{D,z}\}\). The authors study interrelations between pseudoconvexity (resp.~weak linear convexity, resp.~linear convexity) of \(D\), \(H_D\), and \(B_{D,z}\), \(z\in D\). In particular, they prove the following two theorems:NEWLINENEWLINE--- \(D\) is pseudoconvex iff \(H_D\) is pseudoconvex iff \(B_{D,z}\) is pseudoconvex for any \(z\in D\).NEWLINENEWLINE--- (1) \(D\) is weakly linearly convex (resp.~linearly convex) iff (2) \(H_D\) is weakly linearly convex (resp.~linearly convex) \(\Longrightarrow\) (3) \(B_{D,z}\) is convex for any \(z\in D\). Moreover, if \(D\) is \(\mathcal C^{1,1}\)-smooth, then (3) \(\Longrightarrow\) (1).NEWLINENEWLINELet \(D\subset\mathbb C^n\) be an open non-pseudoconvex set, \(n\geq3\). Let \(S\) be the union of all two dimensional complex planes \(P\) such that \(P\cap D\neq\emptyset\) and \(P\cap D\) is non-pseudoconvex. The authors present the following two characterizations of the \textit{exceptional set} \(\mathbb C^n\setminus S\):NEWLINENEWLINE--- There exists a complex hyperplane \(T\) such that \(\mathbb C^n\setminus S\subset T\).NEWLINENEWLINE--- If \(D\) is \(\mathcal C^2\)-smooth then there exists a three-codimensional plane \(T\) such that \(\mathbb C^n\setminus S\subset T\).NEWLINENEWLINEA stronger characterization of the set \(S\) has been given in the paper [\textit{N.~Nikolov} and \textit{P.~Pflug}, Math. Z. 272, No. 1--2, 381--388 (2012; Zbl 1255.32006)].