Mixed Volume Is Symmetric

Prove that mixed volume is symmetric: For any polytopes $P_1,P_2,\dots,P_n \subset \mathbb{R}^n$ and permutation $\sigma$ of $\{1,2,\dots,n\}$, $$ \text{MV}(P_1,P_2,\dots,P_n) = \text{MV}(P_{\sigma(1)}, P_{\sigma(2)}, \dots, P_{\sigma(n)}).$$


Solution:

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