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 $K\subset G$ a compact set, we define the holomorphically convex hull of $K$ as

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

When a function f is assumed to be a polynomial, which is holomorphically everywhere, we say ${\hat {K}}$ is the polynomially 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 ${\hat {K}}_{G}=K$ for all compact $K\subset G.$ Also note that for n=1 being holomorphically convex is the same as being a domain of holomorphy. These concepts are more important in the case n > 1 of several complex variables.

See also

• polynomially convex hull
• Stein manifold

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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