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  

be a domain, that is, an open connected subset. One says that  is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function  on  such that the set  

is a relatively compact subset of  for all  real numbers  In other words, a domain is pseudoconvex if  has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When  has a  (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a   boundary, it can be shown that   has a defining function, i.e., that there exists   which is   so that  , and . Now,   is pseudoconvex iff for every   and  in the complex tangent space at p, that is,  

, we have

The definition above is analogous to definitions of convexity in Real Analysis. If  does not have a  boundary, the following approximation result can be useful. Proposition 1   If   is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains   with  (smooth) boundary which are relatively compact in , such that  

This is because once we have a  as in the definition we can actually find a C∞ exhaustion function. The case n = 1

In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1. See also

 Analytic polyhedron
 Eugenio Elia Levi
 Holomorphically convex hull
 Stein manifold

References

 
 Lars Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1990. (). Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992. External links

 
 

Several complex variables