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