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 talk, we review and extend this idea to the projection body, centroid body, and LYZ Body, as well as prove the associated inequalities (analogues of Zhang's projection inequality, Petty's projection inequality, the Busemann-Petty centroid inequality, Busemann's random simplex inequality and the affine Sobolev inequality).
Joint work with J. Haddad, E. Putterman, M. Roysdon and D. Ye
