Proper Holomorphic Maps in Euclidean Spaces Avoiding Unbounded Convex Sets

We show that if E is a closed convex set in C-n (n > 1) contained in a closed halfspace H such that E boolean AND bH is nonempty and bounded, then the concave domain Omega = C-n\E contains images of proper holomorphicmaps f : X -> 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 <= n then integral 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.