Suppose is an open set (usually connected), then we define
We call the tube domain associated to .
Basically the idea is that these domains give you complete freedom in the imaginary coordinates. An interesting fact about these domains is that is pseudoconvex if and only if is convex in the geometric sense, and this is true if and only if itself is convex.
- 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.