Criterion For Non Zero Mixed Volume

Prove that for polytopes $P_1, P_2, \dots, P_n$ in $\mathbb{R}^n$, $\text{MV}(P_1,P_2,\dots,P_n) > 0$ if and only if there are pairs of vertices $p_i, q_i \in P_i$ for all $i=1,2,\dots,n$ such that the vectors $\{p_i-q_i: 1 \leq i \leq n\}$ are linearly independent.

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