On the $m\mathrm{th}$-Order Weighted Projection Body Operator and Related Inequalities

Abstract

For a convex body $K$ in ${\mathbb{R}}^n$, the inequalities of Rogers-Shephard and Zhang, written succinctly, are $${\text{vol}}_n(DK)\le \left(\genfrac{}{}{0ex}{}{2n}{n}\right){\text{vol}}_n(K)\le {\text{vol}}_n(n{\text{vol}}_n(K){\Pi }^{\circ }K).$$ Here, $DK=\{x\in {\mathbb{R}}^n:K\cap (K+x)\ne \varnothing \}$ is the difference body of $K$, and ${\Pi }^{\circ }K$ is the polar projection body of $K$. There is equality in either if, and only if, $K$ is a $n$-dimensional simplex. In fact, there exists a collection of convex bodies, the so-called radial mean bodies $R_{p}K$ introduced by Gardner and Zhang, which continuously interpolates between $DK$ and ${\Pi }^{\circ }K$. For $m\in \mathbb{N}$, Schneider defined the $m$th-order difference body of $K$ as $$D^m(K)=\{(x_1,\dots ,x_m)\in {\mathbb{R}}^{nm}:K{\cap }_{i=1}^m(K+x_i)\ne \varnothing \}\subset {\mathbb{R}}^{nm}$$ and proved the $m$th-order Rogers-Shephard inequality. In a prequel to this work, the authors, working with Haddad, extended this $m$th-order concept to the radial mean bodies and the polar projection body, establishing the associated Zhang's projection inequality. In this work, we introduce weighted versions of the above-mentioned operators by replacing the Lebesgue measure with measures that have density. The weighted version of these operators in the $m=1$ case was first done by Roysdon (difference body), Langharst-Roysdon-Zvavitch (polar projection body) and Langharst-Putterman (radial mean bodies). This work can be seen as a sequel to all those works, extending them to $m$th-order. In the last section, we extend many of these ideas to the setting of generalized volume, first introduced by Gardner-Hug-Weil-Xing-Ye.

33 pages, Keywords: Projection Bodies, Rogers-Shephard Inequality, Zhang's Inequality, Radial Mean Bodies. Title changed from "higher-order..." to "mth order..." Accepted into Pure and Applied Functional Analysis, Special Issue in honour of Nicole Tomczak-Jaegermann