  In 1970, Schneider generalized the difference body of a convex body to
higher-order, and also established the higher-order analogue of the
Rogers-Shephard inequality. In this paper, we extend this idea to the
projection body, centroid body, and radial mean bodies, as well as prove the
associated inequalities (analogues of Zhang's projection inequality, Petty's
projection inequality, the Busemann-Petty centroid inequality and Busemann's
random simplex inequality). We also establish a new proof of Schneider's
higher-order Rogers-Shephard inequality. As an application, a higher-order
affine Sobolev inequality for functions of bounded variation is provided.
