For a convex body K in ${\mathbb{R}}^n$, we introduce and study the extremal general affine surface areas, defined by${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }\subset K}{\sup }{\mathrm{as}}_{\phi }(K),{\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }\supset K}{\inf }{\mathrm{as}}_{\psi }(K)$ 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.

对于凸体K in${\mathbb{电阻}}^n$，我们介绍和研究极值一般仿射表面积，定义为${\mathrm{\text{是}}}_{\phi }(钾)：=\underset{{钾}^{\prime }\subset 钾}{\mathrm{超}}{\mathrm{作为}}_{\phi }(钾),{\mathrm{\text{操作系统}}}_{\psi }(钾)：=\underset{{钾}^{\prime }\supset 钾}{\mathrm{信息}}{\mathrm{作为}}_{\psi }(钾)$ 在哪里 ${\mathrm{作为}}_{\phi }(钾)$ 和 ${\mathrm{作为}}_{\psi }(钾)$ 是 ${升}_{\phi }$ 和 ${升}_{\psi }$分别为K 的仿射表面积。我们证明存在达到上界和下界的极值凸体，并且泛函${\mathrm{\text{是}}}_{\phi }$ 和 ${\mathrm{\text{操作系统}}}_{\psi }$是连续的。在我们的主要结果中，我们证明了极值一般仿射表面积的 Blaschke-Santaló 型不等式和逆 Santaló 型不等式。这篇文章可以看作是 Giladi、Huang、Schütt 和 Werner (2020) 近期工作的 Orlicz 延伸，他们介绍并研究了极值${升}_{磷}$ 仿射表面积。