# 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

be a domain, that is, an open connected subset. One says that is *pseudoconvex* (or *Hartogs pseudoconvex*) if there exists a continuous plurisubharmonic function on such that the set

is a relatively compact subset of for all real numbers In other words, a domain is pseudoconvex if has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex.

When has a (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a boundary, it can be shown that has a defining function; i.e., that there exists which is so that , and . Now, is pseudoconvex iff for every and in the complex tangent space at p, that is,

- , we have

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

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

This is because once we have a as in the definition we can actually find a *C*^{∞} exhaustion function.

## The case *n* = 1

*n*= 1

In one complex dimension, every open domain is pseudoconvex. The concept of pseudoconvexity is thus more useful in dimensions higher than 1.

## 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.

*This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

## External links

- Range, R. Michael (February 2012), "WHAT IS...a Pseudoconvex Domain?" (PDF),
*Notices of the American Mathematical Society*,**59**(2): 301–303, doi:10.1090/noti798 - Hazewinkel, Michiel, ed. (2001) [1994], "Pseudo-convex and pseudo-concave",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4