Orlicz mixed projection body


Chang-Jian Zhao




In the paper, our main aim is to generalize the mixed projection body Π(K1, . . . ,Kn−1) of (n − 1) convex bodies K1, . . . ,Kn−1 to the Orlicz space. Under the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric operation call it Orlicz mixed projection body Πφ(K1, . . . ,Kn) of n convex bodies K1, . . . ,Kn. The new affine geometric quantity in special case yields the classical mixed projection body Π(K1, . . . ,Kn−1) and Orlicz projection body ΠφK of convex body K, respectively. The related concept of Lp-mixed projection body of n convex bodies Πp(K1, . . . ,Kn) is also derived. An Orlicz Alesandrov- Fenchel inequality for the Orlicz mixed projection body is established, which in special case yields a new Lp-projection Alesandrov-Fenchel inequality. As an application, we establish a polar Orlicz Alesandrov- Fenchel inequality for the polar of Orlicz mixed projection body.