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

For a convex body $K$ in $\mathbb R^n$, the inequalities of Rogers-Shephard and Zhang, written succinctly, are $$\text{vol}_n(DK)\leq \binom{2n}{n} \text{vol}_n(K) \leq \text{vol}_n(n\text{vol}_n(K)Π^\circ K).$$ Here, $DK=\{x\in\mathbb R^n:K\cap(K+x)\neq \emptyset\}$ is the difference body of $K$, and $Π^\circ K$ 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 $Π^\circ K$. For $m\in\mathbb N$, Schneider defined the $m$th-order difference body of $K$ as $$D^m(K)=\{(x_1,\dots,x_m)\in\mathbb R^{nm}:K\cap_{i=1}^m(K+x_i)\neq \emptyset\}\subset \mathbb R^{nm}$$ 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.

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