Characterization of simplices via the Bezout inequality for mixed volumes

We consider the following Bezout inequality for mixed volumes: (Formula Presented). It was shown previously that the inequality is true for any n-dimensional simplex Δ and any convex bodies K1,…, Kr in ℝn. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies K1,…, Kr in ℝn. In this paper we prove that this is indeed the case if we assume that Δ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex n-polytopes. In addition, we show that if a body Δ satisfies the Bezout inequality for all bodies K1,…, Kr, then the boundary of Δ cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.