2025/03/02 by Sola, Francisco Marín
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2503.00949
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.