PlanetMath: logarithmically convex set

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logarithmically convex set

(Definition)

Suppose $G \subset {\mathbb{C}}^n$ , then we define

$\displaystyle \log \lVert G \rVert := \{ (\log \lvert z_1 \rvert ,\ldots, \log \lvert z_n \rvert) \in {\mathbb{R}}^n \mid (z_1,\ldots,z_n) \in G \} .$

Definition 1 We say $G \subset {\mathbb{C}}^n$ is a logarithmically convex set if $\log \lVert G \rVert \subset {\mathbb{R}}^n$ is a convex set.

Bibliography

1: Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

"logarithmically convex set" is owned by jirka.

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Cross-references: convex set
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This is version 2 of logarithmically convex set, born on 2004-07-25, modified 2005-03-07.
Object id is 6027, canonical name is LogarithmicallyConvexSet.
Accessed 1498 times total.

Classification:

AMS MSC:

32A07 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special domains )

Pending Errata and Addenda

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