Oka properties of ball complements

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.