  We establish equivalence between the boundedness of specific supremum
operators and the optimality of function spaces in Sobolev embeddings acting on
domains in ambient Euclidean space with a prescribed isoperimetric behavior.
Our approach is based on exploiting known relations between higher-order
Sobolev embeddings and isoperimetric inequalities. We provide an explicit way
to compute both the optimal domain norm and the optimal target norm in a
Sobolev embedding. Finally, we apply our results to higher-order Sobolev
embeddings on John domains and on domains from the Maz'ya classes. Furthermore,
our results are partially applicable to embeddings involving product
probability spaces.
