Some comments on the mth-order Projection Bodies

TL;DR

This work extends Petty's affine-invariant projection inequality to the mth-order setting and to mixed projection bodies. It defines the $m$th-order polar projection body $\\Pi^{\circ,m}K$ in $\\mathbb{R}^{nm}$ via a variational gauge and proves two sharp bounds: a lower bound $\\frac{1}{n^{nm}} \binom{nm+n}{n}$ attained by simplices and an upper bound involving the mean width $W_n$ attained by ellipsoids. It then develops a mixed version $\\Pi^{\circ,m}(K_1,...,K_{n-1})$ and proves a corresponding mixed-volume inequality with equality iff the bodies are homothetic, plus a corollary bound that ties together the volumes of the $K_i$ and the mixed projection body, with equality when all $K_i$ are ellipsoids. The results are established via affine-invariant reasoning, the Aleksandrov–Fenchel inequality, and a two-lemma framework, enriching the higher-order projection theory with stability-type statements and connections to affine geometric inequalities.

Abstract

The celebrated Petty's projection inequality is a sharp upper bound for the volume of the polar projection body of a convex body. Lutwak introduced the concept of mixed projection bodies and extended Petty's projection inequality. Alonso-Gutiérrez later did a stability result for Petty's projection inequality. In 1970, Schneider introduced the $m$th-order setting and extended the difference body to that setting. In a previous work, we, working with Haddad, Putterman, Roysdon, and Ye, established an extension of the projection body operator to this setting. In this note, we continue this study for the mixed projection body operator as well as the question of stability.

Theorems & Definitions (18)

• Theorem 2.1
• Definition 2.2
• Theorem 2.3: Zhang's projection and Petty's projection inequalities for mth-order projection bodies
• Proposition 2.4: Petty Product for mth-Order Projection Bodies
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• proof
• Lemma 2.8
• proof