On the complements of 3-dimensional convex polyhedra as polynomial images of ${\mathbb{R}}^3$

We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ofa3-dimensionalconvexpolyhedron${\mathcal K}\subset{\mathbb R}^3$ and its closure $\bar{\mathcal{S}}$ are polynomial images of ${\mathbb{R}}^3$. The former techniques cannot be extended in general to represent such semialgebraic sets $\mathcal{S}$ and $\bar{\mathcal{S}}$ as polynomial images of ${\mathbb{R}}^n$ if $n\ge 4$.