# 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.

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 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 (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.