What is "pseudoconvexity"

n. The property of being pseudoconvex.

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 C. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

Let

G ⊂ C

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 φ on G such that the set

{z ∈ G ∣ φ(z) < x}

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.

When G has a C (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C boundary, it can be shown that G has a defining function; i.e., that there exists ρ : C → R which is C so that G = {ρ < 0}, and ∂G = {ρ = 0}. Now, G is pseudoconvex iff for every p ∈ ∂G and w in the complex tangent space at p, that is,

$$\nabla \rho(p) w = \sum_{i=1}^n \frac{\partial \rho (p)}{ \partial z_j }w_j =0$$
, we have

$$\sum_{i,j=1}^n \frac{\partial^2 \rho(p)}{\partial z_i \partial \bar{z_j} } w_i \bar{w_j} \geq 0.$$

If G does not have a C boundary, the following approximation result can come in useful.

Proposition 1 If G is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains G ⊂ G with C ( smooth) boundary which are relatively compact in G, such that

G = ⋃G.

This is because once we have a φ as in the definition we can actually find a C exhaustion function.