# tube domain

###### Definition.

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

$${T}_{S}:=\{z\in {\u2102}^{n}\mid \mathrm{Re}z\in S\}.$$ |

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

Basically the idea is that these domains give you complete^{} freedom in 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 |