The φ-mixed affine surface areas

Abstract

In the paper, our main aim is to introduce a new φ-mixed affine surface area of convex bodies, which obeys classical basic properties. The new affine geometric quantity in special case yields the classical
Lp-affine surface area, Lp-mixed affine surface area  and the newly established Lpq-mixed affine surface area, respectively. As an application, we establish a φ-Minkowski inequality for the φ-mixed affine
surface area, which follows the classical Minkowski inequality for mixed affine surface area, Lp-Minkowski inequality for Lp-affine surface area and Lpq-Minkowski inequality for Lpq-mixed affine surface area, respectively.