Bezout Inequality for Mixed volumes

Abstract

In this paper we consider the following analog of Bezout inequality for mixed volumes: $$V(P_1,\dots,P_r,\Delta^{n-r})V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(P_i,\Delta^{n-1})\ \text{ for }2\leq r\leq n.$$ We show that the above inequality is true when $\Delta$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in $\mathbb{R}^n$. We conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$-dimensional simplex. We prove that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to $\Delta$), which confirms the conjecture when $\Delta$ is a simple polytope and in the 2-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.