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

TL;DR

This paper leverages the $ ext{L}_p$ Busemann-Petty centroid inequality to derive sharp affine functional inequalities such as log-Sobolev, Sobolev, and Gagliardo-Nirenberg inequalities, providing elementary proofs and equality characterizations.

Contribution

It introduces a novel approach using the $ ext{L}_p$ Busemann-Petty centroid inequality to study and establish sharp affine inequalities with geometric significance.

Findings

Derived sharp affine inequalities including log-Sobolev, Sobolev, and Gagliardo-Nirenberg.

Provided elementary proofs for these inequalities.

Characterized the equality cases explicitly.

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.