We show that if E is a closed convex set in $${\mathbb {C}}^n$$
C
n
$$(n>1)$$
(
n
>
1
)
contained in a closed halfspace H such that $$E\cap bH$$
E
∩
b
H
is nonempty and bounded, then the concave domain $$\Omega = {\mathbb {C}}^n{\setminus } E$$
Ω
=
C
n
\
E
contains images of proper holomorphic maps $$f:X\rightarrow {\mathbb {C}}^n$$
f
:
X
→
C
n
from any Stein manifold X of dimension $$<n$$
<
n
, with approximation of a given map on closed compact subsets of X. If in addition $$2\dim X+1\le n$$
2
dim
X
+
1
≤
n
then f can be chosen an embedding, and if $$2\dim X=n$$
2
dim
X
=
n
, then it can be chosen an immersion. Under a stronger condition on E, we also obtain the interpolation property for such maps on closed complex subvarieties.