General measure extensions of projection bodies

The inequalities of Petty and Zhang are affine isoperimetric‐type inequalities providing sharp bounds for Volnn−1(K)Voln(Π∘K)$\text{\rm Vol}^{n-1}_{n}(K)\text{\rm Vol}_n(\Pi ^\circ K)$, where ΠK$\Pi K$ is a projection body of a convex body K$K$. In this paper, we present a number of generalizations...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2022-11, Vol.125 (5), p.1083-1129
Main Authors: Langharst, Dylan, Roysdon, Michael, Zvavitch, Artem
Format: Article
Language:English
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Summary:The inequalities of Petty and Zhang are affine isoperimetric‐type inequalities providing sharp bounds for Volnn−1(K)Voln(Π∘K)$\text{\rm Vol}^{n-1}_{n}(K)\text{\rm Vol}_n(\Pi ^\circ K)$, where ΠK$\Pi K$ is a projection body of a convex body K$K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator Π$\Pi$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang‐type inequalities. Throughout, we apply shown results to the standard Gaussian measure. Under a certain restriction on K$K$, we use our results to obtain a reverse isoperimetric inequality for the measure of ∂K$\partial K$.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms.12477