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 ${\mathcal {O}}(G)$ 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).