Log-concave and spherical models in isoperimetry

Abstract. 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 Lévy—Schmidt theorem on spheres. We use these results to derive a comparison theorem for product measures: the isoperimetric function of $ \mu _1 \otimes\cdots\otimes \mu_n $ is bounded from below in terms of the isoperimetric functions of $ \mu_1,\ldots,\mu_n $. We apply this to measures with finite dimensional isoperimetric behaviors. All the previous estimates can be improved when uniform enlargement is considered.