On the mth-Order Weighted Projection Body Operator and Related Inequalities · arXivDesk
2305.00479Apr 30, 202333 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
On the mth-Order Weighted Projection Body Operator and Related Inequalities
Dylan Langharst, Eli Putterman, Michael Roysdon, Deping Ye
For a convex body K in Rn, the inequalities of Rogers-Shephard and Zhang, written succinctly, are voln(DK)≤(n2n)voln(K)≤voln(nvoln(K)Π∘K).
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Here,
DK={x∈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
RpK
introduced by Gardner and Zhang, which continuously interpolates between
DK
and
Π∘K
. For
m∈N
, Schneider defined the
m
th-order difference body of
K
as
Dm(K)={(x1,…,xm)∈Rnm:K∩i=1m(K+xi)=∅}⊂Rnm
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.