Bezout Inequality for Mixed volumes

TL;DR

This paper investigates a Bezout-type inequality for mixed volumes, proving it for simplices and simple polytopes, and explores its implications for convex geometry and volume inequalities.

Contribution

It establishes the inequality for simplices, proposes a conjecture for all convex bodies, and links the inequality to projection volume inequalities.

Findings

Proved the inequality for simplices and simple polytopes.

Confirmed the conjecture that $ ext{Delta}$ must be indecomposable if inequality holds.

Connected the inequality to projection volume inequalities and provided an isomorphic version.

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.