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Inspired by a recent work of Haddad, Jim\'enez and Montenegro, we give a newand simple approach to the recently established general affine P\'olya-Szeg\"oprinciple. Our approach is based on the general $L_p$ Busemann-Petty centroidinequality and does not rely on the general $L_p$ Petty projection inequalityor the solution of the $L_p$ Minkowski problem. A Brothers-Ziemer-type resultfor the general affine P\'olya-Szeg\"o principle is also established. Asapplications, we reprove some sharp affine Sobolev-type inequalities and settletheir equality conditions. We also prove a stability estimate for the affineSobolev inequality on functions of bounded variation by using our new approach.As a corollary of this stability result, we deduce a stability estimate for theaffine logarithmic--Sobolev inequality.

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