holomorphically convexHolomorphicallyConvex2005-02-22 12:54:352006-04-20 18:34:04Definitionholomorphically convex hull% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}Let $G \subset {\mathbb{C}}^n$ be a domain, or alternatively for a more general definition let $G$ be an $n$ dimensional complex analytic manifold.
Further let ${\mathcal{O}}(G)$ stand for the set of holomorphic functions on $G$.
\begin{defn}
Let $K \subset G$ be a compact set.
We define the {\em holomorphically \PMlinkescapetext{convex hull}} of $K$ as
\begin{equation*}
\hat{K}_G := \{ z \in G \mid \lvert f(z) \rvert \leq \sup_{w \in K} \lvert f(w) \rvert \text{ for all } f \in {\mathcal{O}}(G) \} .
\end{equation*}
The domain $G$ is called {\em holomorphically convex} if for every $K \subset G$ compact in $G$, $\hat{K}_G$ is also compact in $G$. Sometimes this is just abbreviated as {\em holomorph-convex}.
\end{defn}
Note that when $n=1$, any domain $G$ is holomorphically convex since when $n=1$ $\hat{K}_G = K$ for all compact $K \subset G$. Also note that this is the same as being a domain of holomorphy.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}