The φ-mixed volumes

Abstract

In the paper, our main aim is to introduce a new φ-mixed volume of (n+1) convex bodies, which obeys classical properties. The new affine geometric quantity in special case yields the classical mixed volume, p-mixed quermassintegral  and the newly established Lp-multiple mixed volume, respectively. As an application, we establish an Orlicz Alesandrov-Fenchel inequality for the φ-mixed volumes, which follows the classical Alesandrov-Fenchel inequality, Lp-Minkowski inequality for p-mixed quermassintegrals and Lp-Alesandrov-Fenchel inequality, respectively.