In 1970, Schneider introduced the \(m\) th order difference body of a convex body, and also established the \(m\) th-order 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 \(m\) th-order Rogers–Shephard inequality. As an application, a \(m\) th-order affine Sobolev inequality for functions of bounded variation is provided.
