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.
such that the set is a relatively compact subset of
has a continuous plurisubharmonic exhaustion function.
Every (geometrically) convex set is pseudoconvex.
However, there are pseudoconvex domains which are not geometrically convex.
(twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with.
has a defining function, i.e., that there exists
in the complex tangent space at p, that is, The definition above is analogous to definitions of convexity in Real Analysis.
boundary, the following approximation result can be useful.
(smooth) boundary which are relatively compact in
as in the definition we can actually find a C∞ exhaustion function.
In one complex dimension, every open domain is pseudoconvex.
The concept of pseudoconvexity is thus more useful in dimensions higher than 1.
This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.