We start by reviewing the classical Hardy inequality for convex sets. In particu-lar, we recall a couple of classical analytic proofs. We then discuss the counterpart of Hardy’s inequality for the case of fractional Sobolev-Slobodecki˘ı spaces, still in the case of open convex subsets of the Euclidean space. In particular, we determine the sharp constant in this inequality, by constructing explicit supersolutions based on the distance function. We also show that this method works only for the mildly nonlocal regime and it gets stuck for the strongly nonlocal one. We conclude by presenting some open problems.
Some of the results presented are issued from papers in collaboration with Francesca Bianchi (Ferrara & Parma), Eleonora Cinti (Bologna) and Anna Chiara Zagati (Ferrara & Parma).