Higher-order Lp 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 Rnfrom those in Rn, were replaced by inter-dimensional simplicial operators, which generate convex bodies in Rnm 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 Santal & oacute; inequalities, and L p affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals (Rn, center dot E )-* (Rm, center dot F ). (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org /licenses /by /4 .0/).