Characterization of simplices via the Bezout inequality for mixed volumes

Source abstract

We consider the following Bezout inequality for mixed volumes: $$V ( K 1 , … , K r , Δ [ n − r ] ) V n ( Δ ) r − 1 ≤ ∏ i = 1 r V ( K i , Δ [ n − 1 ] ) for 2 ≤ r ≤ n . V(K_1,\dots ,K_r,\Delta [{n-r}])V_n(\Delta )^{r-1} \leq \prod _{i=1}^r V(K_i,\Delta [{n-1}])\ \text { for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any n n -dimensional simplex Δ \Delta and any convex bodies K 1 , … , K r K_1, \dots , K_r in R n \mathbb {R}^n . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies K 1 , … , K r K_1, \dots , K_r in R n \mathbb {R}^n . In this paper we prove that this is indeed the case if we assume that Δ \Delta is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex n n -polytopes. In addition, we show that if a body Δ \Delta satisfies the Bezout inequality for all bodies K 1 , … , K r K_1, \dots , K_r , then the boundary of Δ \Delta cannot have points not lying in a boundary segment. In particular, it cannot have points with positive Gaussian curvature.