The \(L_p\)-generalization of the Petty projection inequality for convex bodies, established by \textit{E.~Lutwak}, \textit{D.~Yang} and \textit{G.~Zhang} [J. Differ. Geom. 56, No. 1, 111--132 (2000; Zbl 1034.52009)], leads to the \(L_p\)-version of the Stam inequality, recently established by the same authors and S.~Lv and called the integral affine Fisher information inequality; see also [\textit{E.~Lutwak}, \textit{D.~Yang} and \textit{G.~Zhang}, IEEE Trans. Inf. Theory 51, No. 2, 473--478 (2005; Zbl 1205.94059)]. By means of the co-area formula, the authors recover the \(L_p\)-Petty projection inequality as a special case of the integral affine Fisher information inequality, provided that \(p\) is smaller than the dimension of the convex body.