Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies

TL;DR

This paper extends classical convex geometric inequalities to higher-order bodies like projection and centroid bodies, providing new proofs and applications in affine Sobolev inequalities for functions of bounded variation.

Contribution

It introduces higher-order versions of key inequalities for projection, centroid, and radial mean bodies, and offers a new proof of Schneider's higher-order Rogers-Shephard inequality.

Findings

Established higher-order affine isoperimetric inequalities.

Proved analogues of Zhang's and Petty's inequalities for higher-order bodies.

Derived a higher-order affine Sobolev inequality for BV functions.

Abstract

In 1970, Schneider introduced the $m$th order difference body of a convex body, and also established the $m$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 $m$th-order Rogers-Shephard inequality. As an application, a $m$th-order affine Sobolev inequality for functions of bounded variation is provided.