2023/04/16 by Julián Haddad, Haddad, Julián, Dylan Langharst +7 · 2 citations
Mathematics · #Point processes and geometric inequalities #Mathematical Inequalities and Applications
paper · pdf · doi:10.48550/arxiv.2304.07859
In 1970, Schneider introduced the mth order difference body of a convex body, and also established the mth-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 mth-order Rogers-Shephard inequality. As an application, a mth-order affine Sobolev inequality for functions of bounded variation is provided.