On the $m$th-order Affine Pólya-Szegö Principle

Abstract: An affine P\'olya-Szeg\"o principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the $L<sup>p$ affine P\'olya-Szeg\"o principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new P\'olya-Szeg\"o principle are shown.