- Show that the mixed volume $MV_n(P_1,\dots,P_n)$ is invariant under all permutations of the $P_i.$
2.Show that the mixed volume is linear in each variable
$MV_n(P_1,\dots,\lambda P_i+\beta P_i',\dots,P_n)=\lambda MV_n(P_1,\dots,P_i,\dots,P_n)+\beta MV_n(P_1,\dots,P_i',\dots,P_n)$ for all $i=1,\dots,n$ and $\lambda, \beta \geq0$ in $\mathbb{R}$.