# tube domain

###### Definition.

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

 $T_{S}:=\{z\in{\mathbb{C}}^{n}\mid\operatorname{Re}z\in S\}.$

We call $T_{S}$ the tube domain associated to $S$.

Basically the idea is that these domains give you complete freedom in the imaginary 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.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title tube domain TubeDomain 2013-03-22 14:32:44 2013-03-22 14:32:44 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 32A07