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 {\displaystyle G\subset {\mathbb {C} }^{n}} be a domain. One says that G {\displaystyle G} is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function φ {\displaystyle \varphi } on G {\displaystyle G} such that the set { z ∈ G ∣ φ ( z ) < x } {\displaystyle \{z\in G\mid \varphi (z)<x\}} is a relatively compact subset of G {\displaystyle G} for all real numbers x . {\displaystyle x.} In other words, a domain is pseudoconvex if G {\displaystyle 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 {\displaystyle G} has a C 2 {\displaystyle 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 {\displaystyle C^{2}} boundary, it can be shown that G {\displaystyle G} has a defining function, i.e., that there exists ρ : C n → R {\displaystyle \rho :\mathbb {C} ^{n}\to \mathbb {R} } which is C 2 {\displaystyle C^{2}} so that G = { ρ < 0 } {\displaystyle G=\{\rho <0\}} , and ∂ G = { ρ = 0 } {\displaystyle \partial G=\{\rho =0\}} . Now, G {\displaystyle G} is pseudoconvex if and only if for every p ∈ ∂ G {\displaystyle p\in \partial G} and w {\displaystyle w} in the complex tangent space at p, that is,
∇ ρ ( p ) w = ∑ i = 1 n ∂ ρ ( p ) ∂ z j w j = 0 {\displaystyle \nabla \rho (p)w=\sum _{i=1}^{n}{\frac {\partial \rho (p)}{\partial z_{j}}}w_{j}=0} , we have
∑ i , j = 1 n ∂ 2 ρ ( p ) ∂ z i ∂ z j ¯ w i w j ¯ ≥ 0. {\displaystyle \sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.}
The definition above is analogous to definitions of convexity in Real Analysis. If G {\displaystyle G} does not have a C 2 {\displaystyle C^{2}} boundary, the following approximation result can be useful. Proposition 1 If G {\displaystyle G} is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains G k ⊂ G {\displaystyle G_{k}\subset G} with C ∞ {\displaystyle C^{\infty }} (smooth) boundary which are relatively compact in G {\displaystyle G} , such that
G = ⋃ k = 1 ∞ G k . {\displaystyle G=\bigcup _{k=1}^{\infty }G_{k}.}
… excerpt ends here. Continue reading the full article.
