  We show that if $E$ is a closed convex set in $\mathbb C^n$ $(n>1)$ contained
in a closed halfspace $H$ such that $E\cap bH$ is nonempty and bounded, then
the concave domain $\Omega = \mathbb C^n\setminus E$ contains images of proper
holomorphic maps $f:X\to \mathbb C^n$ from any Stein manifold $X$ of dimension
$<n$, with approximation of a given map on closed compact subsets of $X$. If in
addition $2\dim X+1\le n$ then $f$ can be chosen an embedding, and if $2\dim
X=n$ 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.
