tube domain

Definition.

Suppose $S\subset {ℝ}^n$ is an open set (usually connected),then we define

$$T_S:=\{z\in {ℂ}^n\mid \mathrm{Re}z\in S\}.$$

We call $T_S$ the tube domain associated to $S$.

Basically the idea is that these domains give you complete freedom in theimaginary coordinates. An interesting fact about these domains is that$T_S$ is pseudoconvex if and only if $T_S$ is convex in the geometric sense,and this is true if and only if $S$ itself is convex.