Higher-order L isoperimetric and Sobolev inequalities

Schneider introduced an inter-dimensional difference body operator on convex bodies, and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in R n from those in R n , were replaced by inter-dimensional simplicial operators, which generate convex bodies in R n m from those in R n (or vice versa). In this work, we treat the L p extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary m -dimensional convex bodies containing the origin. We establish m th-order L p isoperimetric inequalities, including the m th-order versions of the L p Petty projection inequality, L p Busemann-Petty centroid inequality, L p Santaló inequalities, and L p affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals ( R n , ‖ ⋅ ‖ E ) → ( R m , ‖ ⋅ ‖ F ) .