Characterization of Simplices via the Bezout Inequality for Mixed volumes

TL;DR

This paper proves that the Bezout inequality for mixed volumes uniquely characterizes simplices among convex polytopes, and shows bodies satisfying the inequality cannot have points with positive Gaussian curvature.

Contribution

It establishes that simplices are the only convex polytopes satisfying the Bezout inequality for mixed volumes, providing a geometric characterization.

Findings

Simplices are uniquely characterized among convex polytopes by the Bezout inequality.

Bodies satisfying the inequality cannot have points with positive Gaussian curvature.

The boundary of such bodies cannot have strict points.

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 strict points. In particular, it cannot have points with positive Gaussian curvature.