  The class of convex sets that admit approximations as Minkowski sum of a
compact convex set and a closed convex cone in the Hausdorff distance is
introduced. These sets are called approximately Motzkin-decomposable and
generalize the notion of Motzkin-decomposability, i.e. the representation of a
set as the sum of a compact convex set and a closed convex cone. We
characterize these sets in terms of their support functions and show that they
coincide with self-bounded sets, i.e. sets contained in the sum of a compact
convex set and a closed convex cone, if their recession cones are polyhedral
but are more restrictive in general. In particular we prove that a set is
approximately Motzkin-decomposable if and only if its support function has a
closed domain relative to which it is continuous.
