0
$\begingroup$
  1. 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 $\mathbb{R}$.

  • 0
    You might want to give your definition of mixed volume. There are a few, and they are not even the same (they differ by a factor n!), and for some of them those two claims are tautologies.2012-06-19
  • 0
    The $n$ dimensional mixed volume of a collection of polytopes $P_1,\dots,P_n$ denoted by $MV_n(P_1,\dots,P_n)$ is the coefficient of the monomial $\lambda_1 \lambda_2 \dots \lambda_n$ in $vol_n(\lambda_1 P_1+\dots + \lambda_nP_n).$2012-06-19
  • 0
    $vol_n(P)=\frac{1}{n} \sum_F a_F vol'_{n-1}(F)$ where the sum is taken over all facets of $P$. (note $vol'$ is the normalized volume of the facet F of the lattice polytope $P$ given by $vol'_{n-1}(F)=\frac{vol_{n-1}(F)}{vol_{n-1}\mathcal{P}}$, where $\mathcal{P}$ is a fundamental lattice paralletope for $\nu^\perp_F \cap \mathbb{Z}^n.$2012-06-19

1 Answers 1