New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality

Inspired by a recent work of Haddad, Jim\'enez and Montenegro, we give a new and simple approach to the recently established general affine P\'olya-Szeg\"o principle. Our approach is based on the general $L_p$ Busemann-Petty centroid inequality and does not rely on the general $L_p$ Petty projection inequality or the solution of the $L_p$ Minkowski problem. A Brothers-Ziemer-type result for the general affine P\'olya-Szeg\"o 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.