Bezout Inequality for Mixed volumes

In this talk we will discuss 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 will 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 present a 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 will show that the conjecture is true for many special cases, for example, in ${\mathbb R}^2$ or if we assume that $\Delta$ is a convex polytope. Next we will discuss an isomorphic version of the Bezout inequality: what is the best constant $c(n,r)>0$ such that $$ V(P_1,\dots,P_r,\Delta^{n-r})V_n(D)^{r-1}\leq c(n,r)\prod_{i=1}^r V(P_i,D^{n-1})\ \text{ for }2\leq r\leq n, $$ where $P_1, \dots, P_r, D$ are convex bodies in ${\mathbb R}^n.$ Finally, we will present a connection of the above inequality to inequalities on the volume of orthogonal projections of convex bodies as well as inequalities for zonoids.

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