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${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }?K}{\sup }?{\mathrm{as}}_{\phi }(K),{\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }?K}{\inf }?{\mathrm{as}}_{\psi }(K)$ where ${\mathrm{as}}_{\phi }(K)$ and ${\mathrm{as}}_{\psi }(K)$ are the $L_{\phi }$ 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 ${\mathrm{\text{IS}}}_{\phi }$ and ${\mathrm{\text{os}}}_{\psi }$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló 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, Schütt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.