Pseudoconvexity

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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 Cⁿ. 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 G is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function 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 C² (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 with (smooth) boundary which are relatively compact in G, such that

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

[edit] 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.

[edit] See also

• Holomorphically convex hull
• Stein manifold
• Analytic polyhedron

[edit] 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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This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the GFDL.