A brief proof of Bochner's tube theorem and a generalized tube

Abstract: The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M. Abe with giving an example of a finite tube over $\mathbf{C}^n$ for which Abe's theorem no longer holds. The present method may clarify the point where the (affine) convexity comes from.