General Measure Extensions of Projection Bodies

Abstract

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for Vol n−1 n (K)Vol n (Π • K), where ΠK is a projection body of a convex body 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 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 on R n .