Characterization of simplices via the Bezout inequality for mixed volumes

Abstract:

We consider the following Bezout inequality for mixed volumes: \[ V(K_1,\dots ,K_r,\Delta [{n-r}])V_n(\Delta )^{r-1} \leq \prod _{i=1}^r V(K_i,\Delta [{n-1}])\ \text { for }2\leq r\leq n.\] It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ 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 $\Delta$ 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 $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots , K_r$, then the boundary of $\Delta$ cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.