I'm reading Section 1.3 in Santambrogio's Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling.
From the equality between the minimum of $(\mathrm{KP})$ and the maximum of $(\mathrm{DP})$ and the fact that both extremal values are realized, one can consider an optimal transport plan $\gamma$ and a Kantorovich potential $\varphi$ and write $$ \varphi(x)+\varphi^{c}(y) \leq c(x, y) \text { on } \Omega \times \Omega \text { and } \varphi(x)+\varphi^{c}(y)=c(x, y) \text { on } \operatorname{spt}(\gamma) . $$ The equality on $\operatorname{spt}(\gamma)$ is a consequence of the inequality which is valid everywhere and of $$ \int_{\Omega \times \Omega} c \mathrm{~d} \gamma=\int_{\Omega} \varphi \mathrm{d} \mu+\int_{\Omega} \varphi^{c} \mathrm{~d} v=\int_{\Omega \times \Omega}\left(\varphi(x)+\varphi^{c}(y)\right) \mathrm{d} \gamma $$ which implies equality $\gamma$-a.e. These functions being continuous, the equality is satisfied on a closed set, i.e. on the support of the measure $\gamma$ (let us recall the definition of support of a measure, not to be confused with sets where it is concentrated).
Once we have that, let us fix a point $\left(x_{0}, y_{0}\right) \in \operatorname{spt}(\gamma)$. One may deduce from the previous computations that $$ x \mapsto \varphi(x)-c\left(x, y_{0}\right) \quad \textbf{is minimal at} \quad x=x_{0} $$ and, if $\varphi$ and $c\left(\cdot, y_{0}\right)$ are differentiable at $x_{0}$ and $x_{0} \notin \partial \Omega$, one gets $\nabla \varphi\left(x_{0}\right)=\nabla_{x} c\left(x_{0}, y_{0}\right)$. We resume this fact in a very short statement (where we do not put the sharpest assumptions on $c$ ) since we will use it much later on.
My question: We have $\varphi(x) + \varphi^{c}(y) \leq c(x, y)$ for all $(x, y) \in \Omega \times \Omega$. Then $\varphi (x) - c(x, y_0) \le - \varphi^{c}(y_0)$ for all $x \in \Omega$. On the other hand, $\varphi(x_0)+\varphi^{c}(y_0)=c(x_0, y_0)$.
Does the author mean $$ x \mapsto \varphi(x)-c\left(x, y_{0}\right) \quad \text{is maximal at} \quad x=x_{0} $$ rather than $$ x \mapsto \varphi(x)-c\left(x, y_{0}\right) \quad \text{is minimal at} \quad x=x_{0} $$ ?