Old and new author’s results on equivalence of various isoperimetric and isocapacitary inequalities, on one hand, and Sobolev’s type imbedding and compactness theorems, on the other hand, are described. It is proved that
some imbeddings into fractional Besov and Riesz potential spaces are equivalent to isoperimetric inequalities of a new type. It is shown also that Sobolev type inequalities follow from weighted one-dimensional inequalities with the weight expressed in terms of a capacity minimizing function. The proof applies to functions on Riemannian manifolds, so that, for such functions, this result provides a substitute for the rearrangement techniques used to obtain sharp constants in Sobolev inequalities in Rn.