An Asymmetric Affine P\'olya--Szeg\"o Principle

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine P'olya–Szeg"o symmetrization principle for functions on ${\mathbb{R}}^n$. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which is stronger than its classical Euclidean counterpart.