# mixed volume properties

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 $\mathbb{R}$.

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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

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 $$vol_n(\lambda_1P_1 + \dots + \lambda_nP_n) = vol_n(\lambda_{\sigma(1)}P_{\sigma(1)} + \dots + \lambda_{\sigma(n)}P_{\sigma(n)})$$ 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.

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