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

摘要

We show that the Lp 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 characterizedirectly the corresponding equality cases.