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

:$G\subset ^n$

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. 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 $\rho: \mathbb^n \to \mathbb$ which is $C^2$ so that $G=\$, and $\partial G =\$. Now, $G$ is pseudoconvex iff for every $p \in \partial G$ and $w$ in the complex tangent space at p, that is,

:$\nabla \rho(p) w = \sum_^n \fracw_j =0$, we have
:$\sum_^n \frac w_i \bar \geq 0.$

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 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''

:$G = \bigcup_^\infty G_k.$

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.

See also

* Analytic polyhedron
* Eugenio Elia Levi
* Holomorphically convex hull
* Stein manifold

References

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* Lars Hörmander, ''An Introduction to Complex Analysis in Several Variables'', North-Holland, 1990. ().
* Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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External links

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{{Convex analysis and variational analysis

Several complex variables