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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. However, there are pseudoconvex domains which are not geometrically convex.

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

The definition above is analogous to definitions of convexity in Real Analysis.

If does not have a boundary, the following approximation result can be 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.

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

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

- Range, R. Michael (February 2012), "WHAT IS...a Pseudoconvex Domain?" (PDF),
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