Pseudoconvexity

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.

Let

:$Gsubset\; \{mathbb\{C^n$

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 $varphi$ on $G$ such that the set

:

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. Otherwise, the following approximation result can come in useful.

Proposition 1 "If $G$ 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"

:

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.

ee also

* Holomorphically convex hull
* Stein manifold

References

*Lars Hörmander, "An Introduction to Complex Analysis in Several Variables", North-Holland, 1990. (ISBN 0-444-88446-7).
* Steven G. Krantz. "Function Theory of Several Complex Variables", AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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