Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
Abstract
In 1970, Schneider introduced the th order difference body of a convex body, and also established the th-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 th-order Rogers-Shephard inequality. As an application, a th-order affine Sobolev inequality for functions of bounded variation is provided.
Keywords
Cite
@article{arxiv.2304.07859,
title = {Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies},
author = {Julián Haddad and Dylan Langharst and Eli Putterman and Michael Roysdon and Deping Ye},
journal= {arXiv preprint arXiv:2304.07859},
year = {2025}
}
Comments
49 pages Keywords: Projection bodies, Centroid bodies, Radial mean bodies, Busemann-Petty centroid inequality, Steiner symmetrization, Petty projection inequality, affine Sobolev inequality