Orlicz mixed projection body

Abstract

In the paper, our main aim is to generalize the mixed projection body ?(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 ??(K1,...,Kn) of n convex bodies K1,...,Kn. The new affine geometric quantity in special case yields the classical mixed projection body ?(K1,...,Kn?1) and Orlicz projection body ??K of convex body K, respectively. The related concept of Lp-mixed projection body of n convex bodies ?p(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.