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^{n}. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy.

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 $\rho :\mathbb {C} ^{n}\to \mathbb {R}$ which is $C^{2))$ so that $G=\{\rho <0\))$, and $\partial G=\{\rho =0\))$. Now, $G$ is pseudoconvex iff for every $p\in \partial G$ 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

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 1If $G$ is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains $G_{k}\subset G$ with $C^{\infty ))$ (smooth) boundary which are relatively compact in $G$, such that

$G=\bigcup _{k=1}^{\infty }G_{k}.$

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.

Zimmer, Andrew (2019). "Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents". Mathematische Annalen. 374 (3–4): 1811–1844. arXiv:1703.01511. doi:10.1007/s00208-018-1715-7. S2CID253714537.

Fornæss, John; Wold, Erlend (2018). "A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary". Pacific Journal of Mathematics. 297: 79–86. arXiv:1611.04464. doi:10.2140/pjm.2018.297.79. S2CID119149200.