On general versions of the Petty projection inequality

The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter‐dimensional operator, Petty's inequality was generalized in Haddad, Langharst, Putterman, Roysdon, and Ye to the so‐called setting, where is an ‐dimensional compact convex set. In this work, we further extend the Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with ‐concave density (for ). Moreover, when , and motivated by a contemporary empirical reinterpretation of Petty's result by Paouris, Pivovarov, and Tatarko, we explore empirical analogues of this inequality.