  Building off of the a previous work of the current authors, we give a unified
treatment of general higher order $L^p$ isoperimetric inequalities, among which
are the higher order $L^p$ Petty projection inequality, higher order $L^p$
Busemann-Petty centroid inequality, higher order $L^p$ Santal\'o inequalities,
and $L^p$ affine Sobolev inequalities. In this general framework, we give a
family of inequalities that include the general affine isoperimetric and
Sobolev inequalities due to Harbel and Schuster and Lutwak, Yang and Zhang.
