Bezout inequality for mixed volumes

In this article, we consider the following analog of Bezout inequality for mixed volumes: (equation presented) We show that the above inequality is true when δ is an n-dimensional simplex and P1, . . . , Pr are convex bodies in Rn. We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then δ must be an n-dimensional simplex. We prove that if the above inequality is true for all convex bodies P1, . . . , Pr , then δ must be indecomposable (i.e., cannot be written as the Minkowski sum of two convex bodies which are not homothetic to δ), which confirms the conjecture when δ is a simple polytope and in the two-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.