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 $C=(C_{ij})\in\mathbb R^{m\times n}$. Define the transport polytope
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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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For $\varepsilon>0$, define the entropy integrand $h:[0,1]\to\mathbb R$ by $h(x)=x(\log x-1)$ for $x\in(0,1]$ and $h(0)=0$, and define the additive entropic transport value by