Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
  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 $\mathbb{R}$.

share|improve this question
    
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. –  sebigu Jun 19 '12 at 7:30
    
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 $vol_n(\lambda_1 P_1+\dots + \lambda_nP_n).$ –  nour Jun 19 '12 at 10:04
    
$vol_n(P)=\frac{1}{n} \sum_F a_F vol'_{n-1}(F)$ 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 $\nu^\perp_F \cap \mathbb{Z}^n.$ –  nour Jun 19 '12 at 10:09

1 Answer 1

I assume your homework is long overdue. Anyways I guess this could be interesting for somebody else.

Concerning 1): This follows relatively quick from the definition of the mixed volume that you gave in the comment. Note that the Minkowski sum $\lambda_1P_1 + \dots + \lambda_nP_n$ is invariant under permutations. (To prove that, look at the definition of Minkwoski sums and use that $a+b=b+a$ for $a, b \in$ any vector space.) So \begin{equation} vol_n(\lambda_1P_1 + \dots + \lambda_nP_n) = vol_n(\lambda_{\sigma(1)}P_{\sigma(1)} + \dots + \lambda_{\sigma(n)}P_{\sigma(n)}) \end{equation} for any permutation $\sigma$. Hence the coefficient of $\lambda_1\lambda_2\ldots\lambda_n$ of the polynomial on the left equals the coefficient of $\lambda_{\sigma(1)}\lambda_{\sigma(2)}\ldots\lambda_{\sigma(n)}$ on the right hand side.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.