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 Cn is defined as follows.

Let G \subset {\mathbb{C}}^n be a domain (an open and connected set), 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. For a compact set K \subset G, the holomorphically convex hull of K is

 \hat{K}_G := \{ z \in G \big| \left| f(z) \right| \leq \sup_{w \in K} \left| f(w) \right| \mbox{ for all } f \in {\mathcal{O}}(G) \} .

(One obtains a narrower concept of polynomially convex hull by requiring in the above definition that f be a polynomial.)

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.

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

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References

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