Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality

Abstract

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 characterize directly the corresponding equality cases.

16 pages. Corrected version to appear in Journal of Functional Analysis

References in corpus (1)

• General Lp affine isoperimetric inequalities

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• Higher-Order Lp Isoperimetric and Sobolev Inequalities
• A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture
• On the $m$th-order Affine Pólya-Szegö Principle