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 ${\displaystyle \mathbb {C} ^{n}}$ is defined as follows.

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

${\displaystyle {\hat {K}}_{G}:=\left\{z\in G\left||f(z)|\leqslant \sup _{w\in K}|f(w)|{\mbox{ for all }}f\in {\mathcal {O}}(G)\right.\right\}.}$

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

The domain ${\displaystyle G}$ is called holomorphically convex if for every compact subset ${\displaystyle K,{\hat {K}}_{G}}$ is also compact in ${\displaystyle G}$. Sometimes this is just abbreviated as holomorph-convex.

When ${\displaystyle n=1}$, any domain ${\displaystyle G}$ is holomorphically convex since then ${\displaystyle {\hat {K}}_{G}}$ is the union of ${\displaystyle K}$ with the relatively compact components of ${\displaystyle G\setminus K\subset G}$. 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.

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