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

An affine Pólya-Szegö principle for a family of affine energies, with equality condition characterization, is demonstrated. In particular, this recovers, as special cases, the $L^p$ affine Pólya-Szegö principles due to Cianchi, Lutwak, Yang and Zhang, and subsequently Haberl, Schuster and Xiao. Various applications of this new Pólya-Szegö principle are shown.