TL;DR
This paper introduces two new families of general affine surface areas, explores their fundamental properties, and establishes affine isoperimetric inequalities for these and existing $L_{\phi}$ affine surface areas.
Contribution
It presents novel families of affine surface areas and derives key inequalities, expanding the theoretical framework of affine geometry.
Findings
Introduction of two new affine surface area families
Establishment of basic properties and inequalities for these areas
Extension of affine isoperimetric inequalities to $L_{\phi}$ affine surface areas
Abstract
Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for affine surface areas are established.
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