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 ndimensional complex space C^{n}. 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 ntuples 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 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 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 187418 August 1943) was a GermanJewish 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. ...
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 onetoone correspondence with the points on an infinite lineâ€”the number line. ...
When G has a C^{2} (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 with (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. ...
This is because once we have a 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.
References  Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
See also  Levi pseudoconvex
 solution of the Levi problem
 exhaustion function
This article incorporates material from Pseudoconvex on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
