New approach to the affine Pólya–Szegö principle and the stability version of the affine Sobolev inequality

Abstract

Inspired by a recent work of Haddad, Jiménez and Montenegro, we give a new and simple approach to the recently established general affine Pólya–Szegö principle. Our approach is based on the general Lp Busemann–Petty centroid inequality and does not rely on the general Lp Petty projection inequality or the solution of the Lp Minkowski problem. A Brothers–Ziemer-type result for the general affine Pólya–Szegö principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic-Sobolev inequality.