A differential inequality characterizing weak lineal convexity (Q1392388)

Recall that a domain \(\Omega\) in \(\mathbb{C}^n\) with boundary of class \(C^1\) is weakly lineally convex if for any point \(p\in\partial \Omega\), the complex tangent plane of \(\partial\Omega\) at \(p\) does not intersect \(\Omega\). \textit{H. Behnke} and \textit{E. Peschl} [Math. Ann. 111, 158-177 (1935; Zbl 0011.16903)] have proved that if \(\partial\Omega\) is of class at least \(C^2\), weak lineal convexity implies that the real Hessian of a defining function of \(\Omega\) at any point \(p\in\partial\Omega\) takes only nonnegative values on the complex tangent space \(T_p^\mathbb{C}(\partial\Omega)\). In this article, after a brief and clear overview on lineal convexity, the author proves that the converse is true, giving thus an analytic characterization of weak lineal convexity. In the special case of Hartogs domains, an analogous result had been obtained by the author previously [Acta Math. Vietnam. 21, No. 1, 69-94 (1996; Zbl 0889.32003)]. The key idea in the present paper is thus to construct Hartogs domains sharing a tangent plane with \(\partial\Omega\).