Several complex variables

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 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 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 . 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 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. (Wikipedia).

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Related pages

Subset | Eugenio Elia Levi | Plurisubharmonic function | Open set | Stein manifold | Bounded set | Connected space | Mathematics | Domain of holomorphy | Real number | Analytic polyhedron | Friedrich Hartogs | Boundary (topology) | Continuous function | Convex set | Lars Hörmander | Function of several complex variables