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