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

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.