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 taking ${\mathcal {O}}(G)$ 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 $K\subset G$ 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, 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.

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.

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Holomorphically convex hull

See also

References