Characterization of Simplices via the Bezout Inequality for Mixed Volumes

We consider the following Bezout inequality for mixed volumes: V (K1, . . . ,Kr, Δ[n − r])Vn(Δ)r−1 ≤ r i=1 V (Ki, Δ[n − 1]) for 2 ≤ r ≤ n. It was shown previously that the inequality is true for any -dimensional simplex and any convex bodies in . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies in . In this paper we prove that this is indeed the case if we assume that is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex -polytopes. In addition, we show that if a body satisfies the Bezout inequality for all bodies , then the boundary of cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.