In mathematics, more precisely in complex analysis, the holomorphically convex hull of a given compact set in the n-dimensional complex space Cn is defined as follows.
Let
G
⊂
C
n
be a domain (an open and connected set), or alternatively for a more general definition, let
G
be an
n
dimensional complex analytic manifold. Further let
O
(
G
)
stand for the set of holomorphic functions on
G
.
For a compact set
K
⊂
G
, the holomorphically convex hull of
K
is
K
^
G
:=
{
z
∈
G
|
|
f
(
z
)
|
≤
sup
w
∈
K
|
f
(
w
)
|
for all
f
∈
O
(
G
)
}
.
(One obtains a narrower concept of polynomially convex hull by requiring in the above definition that f be a polynomial.)
The domain
G
is called holomorphically convex if for every
K
⊂
G
compact in
G
,
K
^
G
is also compact in
G
. Sometimes this is just abbreviated as holomorph-convex.
When
n
=
1
, any domain
G
is holomorphically convex since then
K
^
G
is the union of
K
with the relatively compact components of
G
∖
K
⊂
G
. Also, being holomorphically convex is the same as being a domain of holomorphy (The Cartan–Thullen theorem). These concepts are more important in the case n > 1 of several complex variables.
