In
mathematics, more precisely in the theory of functions of
several complex variables, a pseudoconvex set is a special type of
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in the ''n''-dimensional complex space C
''n''. Pseudoconvex sets are important, as they allow for classification of
domains of holomorphy.
Let
:
be a domain, that is, an
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
subset. One says that
is ''pseudoconvex'' (or ''
Hartogs pseudoconvex'') if there exists a
continuous plurisubharmonic function In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic ...
on
such that the set
:
is a
relatively compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact.
Properties
Every subset of a compact topological space is relatively compact (sin ...
subset of
for all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s
In other words, a domain is pseudoconvex if
has a continuous plurisubharmonic
exhaustion function. Every (geometrically)
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex.
When
has a
(twice
continuously differentiable)
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
, 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
:
The definition above is analogous to definitions of convexity in Real Analysis.
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
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
) 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.
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.
See also
*
Analytic polyhedron
*
Eugenio Elia Levi
*
Holomorphically convex hull
*
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Ste ...
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
*
*
{{Convex analysis and variational analysis
Several complex variables