  The inequalities of Petty and Zhang are affine isoperimetric-type
inequalities providing sharp bounds for
$\text{vol}^{n-1}_{n}(K)\text{vol}_n(\Pi^\circ K),$ where $\Pi K$ 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 $\Pi$ 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.
