Some connections between isoperimetric and Sobolev-type inequalities - Introduction

For Borel probability measures on metric spaces, we study the interplay be tween isoperimetric and Sobolev-type inequalities and in particular, we explore the question of finding optimal constants via isoperimetric quantities. Also, we give necessary and sufficient conditions for the equivalence between the isoperimetric problem and the validity of some analytic inequalities in Orlicz spaces. Some of these results are then specialized to the real line or the Euclidean space.