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
Canonical name	TubeDomain
Date of creation	2013-03-22 14:32:44
Last modified on	2013-03-22 14:32:44
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 32A07