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We consider the following Bezout inequality for mixed volumes:$$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^rV(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously thatthe inequality is true for any $n$-dimensional simplex $\Delta$ and any convexbodies $K_1, \dots, K_r$ in $\mathbb{R}^n$. It was conjectured that simplicesare the only convex bodies for which the inequality holds for arbitrary bodies$K_1, \dots, K_r$ in $\mathbb{R}^n$. In this paper we prove that this is indeedthe case if we assume that $\Delta$ is a convex polytope. Thus the Bezoutinequality characterizes simplices in the class of convex $n$-polytopes. Inaddition, we show that if a body $\Delta$ satisfies the Bezout inequality forall bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strictpoints. In particular, it cannot have points with positive Gaussian curvature.

Characterization of Simplices via the Bezout Inequality for Mixed volumes