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arXiv · 1512.05442

Characterization of Simplices via the Bezout Inequality for Mixed volumes

Christos Saroglou·Ivan Soprunov·Artem Zvavitch
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Abstract

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

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Christos Saroglou, Ivan Soprunov, Artem Zvavitch. 2015-12-17. Characterization of Simplices via the Bezout Inequality for Mixed volumes. https://doi.org/10.1090/proc%2F13149

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