  We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp
Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n
and in the half-space R^n\_+. This gives a new bridge between the geometric
pont of view of the Brunn-Minkowski inequality and the functional point of view
of the Sobolev type inequalities. In this way we unify, simplify and results by
S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret.
