  Recently, Giladi, Huang, Sch\"utt and Werner (2020) introduced and studied
the extremal $L_p$ affine surface areas. In this article, we extend their
definitions to the general $L_\varphi$ and $L_\psi$ affine surface areas, where
$\varphi$ and $\psi$ are concave and convex functions, respectively. We extend
some of their results from the setting of $p$-affine surface areas to the
setting of general affine surface areas, for concave and convex functions
satisfying some prescribed conditions. Finally, we prove Blaschke-Santal\'o
inequalities for these new extremal affine surface areas.
