Let $m,n\in\mathbb N$. Let $a=(a_1,\dots,a_m)\in(0,\infty)^m$ and $b=(b_1,\dots,b_n)\in(0,\infty)^n$ satisfy
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\begin{align*}
\sum_{i=1}^m a_i=1
\end{align*}
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and
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\begin{align*}
\sum_{j=1}^n b_j=1.
\end{align*}
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Let $K=(K_{ij})\in(0,\infty)^{m\times n}$. Define the transport polytope with marginals $a$ and $b$ by
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\begin{align*}
\Pi(a,b):=\left\{P\in[0,\infty)^{m\times n}: \sum_{j=1}^n P_{ij}=a_i \text{ for every } i\in\{1,\dots,m\},\ \sum_{i=1}^m P_{ij}=b_j \text{ for every } j\in\{1,\dots,n\}\right\}.
\end{align*}
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Then there exist vectors $u=(u_1,\dots,u_m)\in(0,\infty)^m$ and $v=(v_1,\dots,v_n)\in(0,\infty)^n$ such that the matrix
belongs to $\Pi(a,b)$. Moreover, if $\widetilde u\in(0,\infty)^m$ and $\widetilde v\in(0,\infty)^n$ also satisfy $\operatorname{diag}(\widetilde u)K\operatorname{diag}(\widetilde v)\in\Pi(a,b)$, then there exists $\lambda>0$ such that
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\begin{align*}
\widetilde u=\lambda u
\end{align*}