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Encyclopedia > 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

Let

$Gsubset {mathbb{C}}^n$

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 In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a connected space is a topological space which cannot be written as the disjoint union of two or more nonempty spaces. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... Friedrich Moritz Hartogs (20 May 1874-18 August 1943) was a German-Jewish mathematician, known for work on set theory and foundational results on several complex variables. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

${ z in G mid varphi(z) < x }$

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. In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

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. In mathematics, a smooth function is one that is infinitely differentiable, i. ... In topology, the boundary of a subset S of a topological space X is the sets closure minus its interior. ...

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 In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

$G = bigcup_{k=1}^infty G_k.$

This is because once we have a $varphi$ 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.

## References

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

• Levi pseudoconvex
• solution of the Levi problem
• exhaustion function

Results from FactBites:

 PlanetMath: pseudoconvex (200 words) The reason for the definition of pseudoconvexity is that it classifies domains of holomorphy. This is version 2 of pseudoconvex, born on 2004-08-02, modified 2005-03-07. Object id is 6056, canonical name is Pseudoconvex.
 PlanetMath: Levi pseudoconvex (278 words) Note that if a point is not strongly Levi pseudoconvex then it is sometimes called a weakly Levi pseudoconvex point. Note that in particular all convex domains are pseudoconvex. This is version 6 of Levi pseudoconvex, born on 2004-07-29, modified 2006-06-21.
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