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 ${\mathbb{R}}^n$ from those in ${\mathbb{R}}^n$, were replaced by inter-dimensional simplicial operators, which generate convex bodies in ${\mathbb{R}}^{nm}$ from those in ${\mathbb{R}}^n$ (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 $({\mathbb{R}}^n,\Vert \cdot {\Vert }_E)\to ({\mathbb{R}}^m,\Vert \cdot {\Vert }_F)$.

39 pgs, results in Santalo section expanded; expanded literature review, expanded section on LYZ body, and changed intro and abstract