Characterization of Simplices via the Bezout Inequality for Mixed volumes

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots ,K_r,\mathrm{\Delta }[n-r])V_n(\mathrm{\Delta }{)}^{r-1}\le \prod _{i=1}^{r}V(K_i,\mathrm{\Delta }[n-1])\text{ for }2\le r\le n\mathrm{.}$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\mathrm{\Delta }$ and any convex bodies $K_1,\dots ,K_r$ in ${\mathbb{R}}^n$. It was conjectured that simplices are 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 indeed the case if we assume that $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ satisfies the Bezout inequality for all bodies $K_1,\dots ,K_r$ then the boundary of $\mathrm{\Delta }$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.