  Optimal higher-order Sobolev type embeddings are shown to follow via
isoperimetric inequalities. This establishes a higher-order analogue of a
well-known link between first-order Sobolev embeddings and isoperimetric
inequalities. Sobolev type inequalities of any order, involving arbitrary
rearrangement-invariant norms, on open sets in $\rn$, possibly endowed with a
measure density, are reduced to much simpler one-dimensional inequalities for
suitable integral operators depending on the isoperimetric function of the
relevant sets.
  As a consequence, the optimal target space in the relevant Sobolev embeddings
can be determined both in standard and in non-standard classes of function
spaces and underlying measure spaces. In particular, our results are applied to
any-order Sobolev embeddings in regular (John) domains of the Euclidean space,
in Maz'ya classes of (possibly irregular) Euclidean domains described in terms
of their isoperimetric function, and in families of product probability spaces,
of which the Gauss space is a classical instance.
