# Holomorphically convex hull

In mathematics, more precisely in complex analysis, the **holomorphically convex hull** of a given compact set in the *n*-dimensional complex space [math]\Complex^n[/math] is defined as follows.

Let [math]G \subset \Complex^n[/math] be a domain (an open and connected set), or alternatively for a more general definition, let [math]G[/math] be an [math]n[/math] dimensional complex analytic manifold. Further let [math]{\mathcal{O}}(G)[/math] stand for the set of holomorphic functions on [math]G.[/math] For a compact set [math]K \subset G[/math], the **holomorphically convex hull** of [math]K[/math] is

- [math] \hat{K}_G := \left \{ z \in G \left | |f(z)| \leqslant \sup_{w \in K} |f(w)| \mbox{ for all } f \in {\mathcal{O}}(G) \right. \right \} .[/math]

One obtains a narrower concept of **polynomially convex hull** by taking [math]\mathcal O(G)[/math] instead to be the set of complex-valued polynomial functions on *G*. The polynomially convex hull contains the holomorphically convex hull.

The domain [math]G[/math] is called **holomorphically convex** if for every compact subset [math]K, \hat{K}_G[/math] is also compact in [math]G[/math]. Sometimes this is just abbreviated as *holomorph-convex*.

When [math]n=1[/math], any domain [math]G[/math] is holomorphically convex since then [math]\hat{K}_G[/math] is the union of [math]K[/math] with the relatively compact components of [math]G \setminus K \subset G[/math]. Also, being holomorphically convex is the same as being a domain of holomorphy (The Cartan–Thullen theorem). These concepts are more important in the case of several complex variables (*n* > 1).

## See also[edit]

## References[edit]

- Lars Hörmander.
*An Introduction to Complex Analysis in Several Variables*, North-Holland Publishing Company, New York, New York, 1973. - Steven G. Krantz.
*Function Theory of Several Complex Variables*, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

*This article incorporates material from Holomorphically convex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

*https://en.wikipedia.org/wiki/Holomorphically convex hull was the original source. Read more*.