Two-dimensional slices of non-pseudoconvex open sets

Abstract

Let $D$ be a non-pseudoconvex open set in $\C^3$ and $S$ be the union of all two-dimensional planes with non-empty and non-pseudoconvex intersection with $D.$ Sufficient conditions are given for $\C^3\setminus S$ to belong to a complex line. Moreover, in the $\mathcal C^2$-smooth case, it is shown that $S=\C^n$.