Sharp affine Sobolev type inequalities via the Busemann-Petty centroid inequality
Abstract
We show that the 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.
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