Pseudoconvexity

In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the n-dimensional complex space Cn. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

Let G\subset {\mathbb{C}}^n be a domain. One says that G is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function \varphi on G such that the set \{ z \in G \mid \varphi(z) is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.