  Poincar\'{e}-Sobolev-type inequalities involving rearrangement-invariant
norms on the entire $\mathbb{R}^n$ are provided. Namely, inequalities of the
type $\|u-P\|_{Y(\mathbb{R}^n)}\leq C\|\nabla^m u\|_{X(\mathbb{R}^n)}$, where
$X$ and $Y$ are either rearrangement-invariant spaces over $\mathbb{R}^n$ or
Orlicz spaces over $\mathbb{R}^n$, $u$ is a $m-$times weakly differentiable
function whose gradient is in $X$, $P$ is a polynomial of order at most $m-1$,
depending on $u$, and $C$ is a constant independent of $u$, are studied. In a
sense optimal rearrangement-invariant spaces or Orlicz spaces $Y$ in these
inequalities when the space $X$ is fixed are found. A variety of particular
examples for customary function spaces are also provided.
