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Let $P,Q$ be polytopes in $\mathbb{R}^n$. If $\lambda,\beta \geq 0$ are in $\mathbb{R}$, show that the $vol_n(\lambda P+ \beta Q)$ can be expressed in terms of mixed volumes as follows:

$\frac{1}{n!} \displaystyle \sum_{k=0}^n {n \choose k} \lambda^k \beta^{n-k} MV_n(P,\dots,P,Q,\dots,Q),$ where in the term corresponding to $k$, $P$ is repeated $k$ times and $Q$ is repeated $n-k$ times in the mixed volume.

(use $n! vol_n(\lambda P+ \beta Q)=MV_n(\lambda P+\beta Q,\dots,\lambda P+\beta Q).)$

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    Instead of posting multiple problems that appear to be homework exercises, you'll probably get a better response if you post a single problem with an attempt made at the solution and an indication where you are stuck.2012-06-18
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    I proved that $MV_n(P,\dots,P)=n!vol_n(P)$2012-06-19

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