Log-concave and spherical models in isoperimetry

We derive several functional forms of isoperimetric inequalities, in the case, of concave isoperimetric profile. In particular, we answer the question of a canonical and sharp functional form of the Levy-Schmidt theorem on spheres. We use these results to derive a comparison theorem for product measures: the isoperimetric function of mu(1)circle times...circle timesmu(n) is bounded from below in terms of the isoperimetric functions of mu(1),...mu(n). We apply this to measures with finite dimensional isoperimetric behaviors. All the previous estimates can be improved when uniform enlargement is considered.