On the mth-Order Weighted Projection Body Operator and Related Inequalities

For a convex body K in Rⁿ, the inequalities of Rogers-Shephard and Zhang, written succinctly, are volₙ(DK)≤ 2nn volₙ(K) ≤ volₙ(nvolₙ(K)Π^∘ K). Here, DK=∈ Rⁿ:K∩(K+x)≠ ∅\ is the difference body of K, and Π^∘ 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 Rₚ K introduced by Gardner and Zhang, which continuously interpolates between $DK$ and Π^∘ K. For m∈ N, Schneider defined the mth-order difference body of K as Dᵐ(K)=\(x₁,,xₘ)∈ Rⁿᵐ:K∩ᵢ₌₁ᵐ(K+xᵢ)≠ ∅\⊂ Rⁿᵐ and proved the mth-order Rogers-Shephard inequality. In a prequel to this work, the authors, working with Haddad, extended this mth-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 mth-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.