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
G
⊂
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
φ
on
G
such that the set
{
z
∈
G
∣
φ
(
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. Every (geometrically) convex set is pseudoconvex.
When
G
has a
C
2
(twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a
C
2
boundary, it can be shown that
G
has a defining function; i.e., that there exists
ρ
:
C
n
→
R
which is
C
2
so that
G
=
{
ρ
<
0
}
, and
∂
G
=
{
ρ
=
0
}
. Now,
G
is pseudoconvex iff for every
p
∈
∂
G
and
w
in the complex tangent space at p, that is,
∇
ρ
(
p
)
w
=
∑
i
=
1
n
∂
ρ
(
p
)
∂
z
j
w
j
=
0
, we have
∑
i
,
j
=
1
n
∂
2
ρ
(
p
)
∂
z
i
∂
z
j
¯
w
i
w
j
¯
≥
0.
If
G
does not have a
C
2
boundary, the following approximation result can come in useful.
Proposition 1 If
G
is pseudoconvex, then there exist bounded, strongly Levi pseudoconvex domains
G
k
⊂
G
with
C
∞
(smooth) boundary which are relatively compact in
G
, such that
G
=
⋃
k
=
1
∞
G
k
.
This is because once we have a
φ
as in the definition we can actually find a C∞ exhaustion function.
