Sharp affine Sobolev type inequalities via the Lp Busemann–Petty centroid inequality

We show that the LpLp Busemann–Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo–Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.