  For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal
general affine surface areas, defined by \[ {\rm
IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad {\rm
os}_{\psi}(K):=\inf_{K^\prime\supset K}{\rm as}_{\psi}(K) \] where ${\rm
as}_{\varphi}(K)$ and ${\rm as}_{\psi}(K)$ are the $L_\varphi$ and $L_\psi$
affine surface area of $K$, respectively. We prove that there exist extremal
convex bodies that achieve the supremum and infimum, and that the functionals
${\rm IS}_{\varphi}$ and ${\rm os}_{\psi}$ are continuous. In our main results,
we prove Blaschke-Santal\'o type inequalities and inverse Santal\'o type
inequalities for the extremal general affine surface areas. This article may be
regarded as an Orlicz extension of the recent work of Giladi, Huang, Sch\"utt
and Werner (2020), who introduced and studied the extremal $L_p$ affine surface
areas.
