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 to the so-called (Lₚ, Q) setting, where Q is an m-dimensional compact convex set. In this work, we further extend the (Lₚ, Q) Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with γ-concave density (for γ-1/nm). Moreover, when p=1, and motivated by a contemporary empirical reinterpretation of Petty's result, we explore empirical analogues of this inequality.