Let $K$ be a convex body in $\mathbb R^n$ (i.e. a set that is compact and convex). Is it possible to construct a domain with smooth boundaries that contains $K$ and is close enough in the sense of Hausdorff distance?
Convex bodies can be approximated by polyhedra from the outside, so I think it's okay if polyhedra can be approximated by smooth domain.
Try this:
There is a one-one correspondence between compact convex sets containing $0$ in its interior, and gauges. A gauge is a continuous function $g:\mathbb R^n \to [0,\infty)$, almost satisfying the axioms of a norm, viz $$ g(x) = 0 \Leftrightarrow x = 0, \quad g(\lambda x) = \lambda g(x) \text{ for } \lambda \ge 0, \quad g(x+y) \le g(x) + g(y) .$$ The correspondence is given by $$ K \mapsto g(x) = \inf\{\lambda : x / \lambda \in K \} ,$$ $$ g \mapsto K = \{ x : g(x) \le 1 \} .$$ Then the issue becomes approximating a gauge by a smooth gauge, which is much easier.