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 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 convex set is pseudoconvex.
When has a 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,
If does not have a boundary, the following approximation result can come in useful.
Proposition 1 If is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with 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.
