holomorphically convex

Let Gn be a domain, or alternatively for a more general definition let G be an n dimensional complex analytic manifold. Further let 𝒪(G) stand for the set of holomorphic functionsMathworldPlanetmath on G.


Let KG be a compact set. We define the holomorphically of K as

K^G:={zG|f(z)|supwK|f(w)| for all f𝒪(G)}.

The domain G is called holomorphically convex if for every KG compact in G, K^G is also compact in G. Sometimes this is just abbreviated as holomorph-convex.

Note that when n=1, any domain G is holomorphically convex since when n=1 K^G=K for all compact KG. Also note that this is the same as being a domain of holomorphy.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title holomorphically convex
Canonical name HolomorphicallyConvex
Date of creation 2013-03-22 15:04:33
Last modified on 2013-03-22 15:04:33
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 32E05
Synonym holomorph-convex
Related topic PolynomiallyConvexHull
Related topic SteinManifold
Defines holomorphically convex hull