The φ-mixed volumes

In the paper, our main aim is to introduce a new \varphi -mixed volume \bar{V}_{\varphi}(K_1, \dots, K_n, L_n) of (n+1) convex bodies, which obeys classical properties. The new affine geometric quantity in special case yields the classical mixed volume V(K_1, \dots, K_n) , p -mixed quermassintegral W_{p,i}(K, L) and the newly established L_p -multiple mixed volume V_{\varphi}(K_1, \dots, K_n, L_n) , respectively. As an application, we establish an Orlicz Alesandrov-Fenchel inequality for the \varphi -mixed volumes, which follows the classical Alesandrov-Fenchel inequality, L_p -Minkowski inequality for p -mixed quermassintegrals and L_p -Alesandrov-Fenchel inequality, respectively.