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

[math]\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 \} . }[/math]

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

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

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