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
G ⊂ C n 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 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 ρ : C n → R which is C 2 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,
∇ ρ ( p ) w = ∑ i = 1 n ∂ ρ ( p ) ∂ z j w j = 0 , we have
∑ i , j = 1 n ∂ 2 ρ ( p ) ∂ z i ∂ z j ¯ w i w j ¯ ≥ 0. If G does not have a C 2 boundary, the following approximation result can come in useful.
Proposition 1 If G is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains G k ⊂ G with C ∞ (smooth) boundary which are relatively compact in G , such that
G = ⋃ k = 1 ∞ G k . This is because once we have a φ as in the definition we can actually find a C∞ exhaustion function.
