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.

be a domain is pseudoconvex (or Hartogs pseudoconvex) if there exists a continuous plurisubharmonic function is a relatively compact for all real numbers has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.

(twice continuously differentiable) boundary in the complex tangent space at p, that is,

1. , we have

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

boundary, the following approximation result can be useful.

Proposition 1 is pseudoconvex, then there exist bounded (smooth, such that

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.

See also

• Analytic polyhedron

• Eugenio Elia Levi

• Holomorphically convex hull

• Stein manifold

References

• 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.

External links