Zhang's inequality for log-concave functions

Zhang’s reverse affine isoperimetric inequality states that among all convex bodies $K\subseteq {\mathbb{R}}^n$, the affine invariant quantity $\mathrm{\mid }K{\mathrm{\mid }}^{n-1}\mathrm{\mid }{\mathrm{\Pi }}^{\ast }(K)\mathrm{\mid }$ (where ${\mathrm{\Pi }}^{\ast }(K)$ denotes the polar projection body of $K$) is minimized if and only if $K$ is a simplex. In this paper we prove an extension of Zhang’s inequality in the setting of integrable log-concave functions, characterizing also the equality cases.