Extremal general affine surface areas

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