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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. –  Michael Joyce Jun 18 '12 at 22:59
    
I proved that $MV_n(P,\dots,P)=n!vol_n(P)$ –  nour Jun 19 '12 at 10:50

1 Answer 1

This follows from the fact that the mixed volume is linear in each argument, i.e. \begin{equation} MV(P_1,\ldots,\lambda P_i + \mu Q_i, \ldots, P_n) = \lambda MV(P_1,\ldots, P_i, \ldots, P_n) + \mu MV(P_1,\ldots, Q_i, \ldots, P_n) \end{equation} Use this to expand $MV_n(\lambda P + \beta Q,\ldots,\lambda P + \beta Q)$ and then collect similar terms.

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