Higher-order Lp isoperimetric and Sobolev inequalities

Abstract

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 $R^{nm}$ 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 mth-order $L^p$ isoperimetric inequalities, including the mth-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 $(R^n,{\Vert \cdot \Vert }_E)\to (R^m,{\Vert \cdot \Vert }_F)$.