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.