Affine isoperimetric inequalities

The sharp Lp Sobolev inequality on Euclidean space by Aubin and Talenti and the associated Polya-Szego principle are famed inequalities in functional analysis. In a series of works, Cianchi, Lutwak, Yang, and Zhang established sharper, affine invariant, versions of these inequalities using new results from convex geometry: Lp extensions of the Busemann-Petty Centroid and Petty projection inequalities. This was quickly followed by Haberl and Schuster, who developed asymmetric versions of these results. In this work, motivated by a conjectured multi-dimensional isoperimetric inequality for convex bodies by Schneider, we establish mth-order (i.e. involving R^n and R^{nm}) extensions of the aforementioned results.