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 G is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function \varphi on G such that the set 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. When G has a C^2 (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C^2 boundary, it can be shown that G has a defining function, i.e., that there exists which is C^2 so that , and. Now, G is pseudoconvex iff for every and w in the complex tangent space at p, that is, The definition above is analogous to definitions of convexity in Real Analysis. If G does not have a C^2 boundary, the following approximation result can be useful. Proposition 1 If G is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains with C^\infty (smooth) boundary which are relatively compact in G, such that This is because once we have a \varphi 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.