# 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 $\Complex^n$ is defined as follows.

Let $G \subset \Complex^n$ 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

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

One obtains a narrower concept of polynomially convex hull by taking $\mathcal O(G)$ 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 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).