Let $X$ and $Y$ be sets, let $c:X \times Y \to [-\infty,\infty]$ be an extended-real-valued cost function, and interpret all infima in $[-\infty,\infty]$. For a function $\varphi:X \to [-\infty,\infty]$, define its $c$-transform $\varphi^c:Y \to [-\infty,\infty]$ by
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\begin{align*}
\varphi^c(y) := \inf_{x \in X} \bigl(c(x,y)-\varphi(x)\bigr), \qquad y \in Y,
\end{align*}
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whenever $c(x,y)-\varphi(x)$ is defined for every $x \in X$ and $y \in Y$. For a function $\psi:Y \to [-\infty,\infty]$, define its $c$-transform $\psi^c:X \to [-\infty,\infty]$ by
whenever $c(x,y)-\psi(y)$ is defined for every $x \in X$ and $y \in Y$.
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If $\varphi:X \to [-\infty,\infty]$ satisfies that, for every $x \in X$ and $y \in Y$, both differences $c(x,y)-\varphi(x)$ and $c(x,y)-\varphi^c(y)$ are defined, equivalently neither is an undefined extended-real form $\infty-\infty$ or $-\infty-(-\infty)$, then
If, in addition, $\varphi$ is $c$-concave in the following explicit sense: there exists a function $\psi:Y \to [-\infty,\infty]$ such that $\varphi=\psi^c$ and, for every $x \in X$ and $y \in Y$, the three differences $c(x,y)-\psi(y)$, $c(x,y)-\psi^c(x)$, and $c(x,y)-\psi^{cc}(y)$ are defined, where