On the monotone properties of general affine surfaces under the Steiner symmetrization Deping Ye 52A2053A15affine surface area$L_p$ Brunn-Minkowski theoryaffine isoperimetric inequalitySteiner symmetrizationthe Orlicz-Brunn-Minkowski theory In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ affine surface area is monotone increasing, while the $L_{\psi}$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5205 10.1512/iumj.2014.63.5205 en Indiana Univ. Math. J. 63 (2014) 1 - 19 state-of-the-art mathematics http://iumj.org/access/