  Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds
$X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact
convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with
approximation and interpolation. If $L$ is polynomially convex then the same
holds when $2\dim X < n$. We also construct proper holomorphic maps, immersions
and embeddings $X\to\mathbb{C}^n$ with additional control of the range, thereby
extending classical results of Remmert, Bishop and Narasimhan.
