# holomorphically convex

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 $G$.

###### Definition.

Let $K\subset G$ be a compact set. We define the holomorphically of $K$ as

 $\hat{K}_{G}:=\{z\in G\mid\lvert f(z)\rvert\leq\sup_{w\in K}\lvert f(w)\rvert% \text{ for all }f\in{\mathcal{O}}(G)\}.$

The domain $G$ is called 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 holomorph-convex.

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.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title holomorphically convex HolomorphicallyConvex 2013-03-22 15:04:33 2013-03-22 15:04:33 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 32E05 holomorph-convex PolynomiallyConvexHull SteinManifold holomorphically convex hull