A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities

We show that mass transportation methods provide an elementary and powerful approach to the study of certain functional inequalities with a geometric content, like sharp Sobolev or Gagliardo-Nirenberg inequalities. The Euclidean structure of R-n plays no role in our approach: we establish these inequalities, together with cases of equality, for an arbitrary norm. (C) 2003 Elsevier Inc. All rights reserved.