Inequalities for general mixed affine surface areas

TL;DR

This paper introduces new general mixed affine surface areas, proves their key properties like affine invariance, and establishes several important inequalities including Alexandrov-Fenchel, Santaló-type, and affine isoperimetric inequalities.

Contribution

It presents novel definitions of mixed affine surface areas and proves significant inequalities and properties that extend classical affine geometry results.

Findings

Established affine invariance of the new surface areas

Proved Alexandrov-Fenchel type inequalities

Derived Santaló-type and affine isoperimetric inequalities

Abstract

Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities, Santal\'{o}-type inequalities, and affine isoperimetric inequalities for these general mixed affine surface areas.