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 ${\mathbb{ℂ}}^n$ 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 ${𝒪}(G)$ stand for the set of holomorphic functions on $G.$ For a compact set $K\subset G$, the holomorphically convex hull of $K$ is

${\hat{K}}_G:=\{z\in G||f(z)|⩽\underset{w\in K}{\sup }|f(w)|\text{ for all }f\in {𝒪}(G)\}.$

One obtains a narrower concept of polynomially convex hull by taking ${𝒪}(G)$ instead to be the set of complex-valued polynomial functions on G. The polynomially convex hull contains the holomorphically convex hull.

The domain $G$ is called holomorphically convex if for every compact subset $K,{\hat{K}}_G$ 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, 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

• Stein manifold
• Pseudoconvexity

References

• 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.