I'm reading the paper Convergence of Entropic Schemes for Optimal Transport and Gradient Flows in which I suspect one argument in Lemma 2.11 is wrong. First, I start with the notation used in the paper.
$\mathcal X = \mathbb R^d$ and $c$ is the Euclidian distance on $\mathcal X$.
$\mathcal P_2(\mathcal X)$ be the set of Borel probability measures on $\mathcal X$ with finite second moment.
$\mu,\nu \in \mathcal P_2(\mathcal X)$.
$\Pi(\mu,\nu)$ is the set of all probability measures on $\mathcal X^2$ with marginals $\mu$ on the first $\mathcal X$ and $\nu$ on the second $\mathcal X$.
We write $(c, \gamma)$ for $\int_{\mathcal X^2} c(x,y) \, \mathrm d \gamma(x,y)$ and define $W^2 (\mu, \nu) = \inf_{\gamma \in \Pi(\mu,\nu)} (c, \gamma)$.
Then I state the Lemma 2.11.
Let $(Q_i)_{i \in I}$ be a countable partition of $\mathcal X$ into Borel sets with $\sup _{i \in I} \operatorname{diam}\left(Q_{i}\right) \leq C< +\infty$, i.e., $\|x-y\|^{2} \leq C^{2}$ for any $x, y \in Q_{i}$ for any $i \in I$. Let $\mu,\nu \in \mathcal P_2 (\mathcal X)$ such that $\mu(Q_i) = \nu(Q_i)$ for all $i \in I$. Then $W^{2}(\mu, \nu) \leq C^{2}$.
Below is the proof provided in the paper:
Denote by $\hat I$ the subset of $I$ such that $\mu\left(Q_{i}\right) = \nu\left(Q_{i}\right)>0$ for $i \in \hat {I}$. For $i \in \hat I$ and every Borel $\sigma \subset \mathcal X$ let
$$ \mu_{i}(\sigma)=\frac{\mu\left(\sigma \cap Q_{i}\right)}{\mu\left(Q_{i}\right)} $$
and analogously define $\nu_{i} .$ Clearly all $\mu_{i}, \nu_{i} \in \mathcal{P}_{2}\left(\mathcal X\right)$, with support contained in $\overline{Q_{i}}$. For every $i \in \hat{I}$ let $\gamma_{i} \in \Pi\left(\mu_{i}, \nu_{i}\right)$ and have spt $\gamma_{i} \subset \overline{Q_{i}}^{2}$ and thus
$$ \left(c, \gamma_{i}\right)=\int_{\mathcal X^2}\|x-y\|^{2} \, \mathrm d \gamma_{i}(x, y)=\int_{Q_{i} \times Q_{i}}\|x-y\|^{2} \, \mathrm d \gamma_{i}(x, y) \leq C^{2} $$
One finds $\color{blue}{\gamma=\sum_{i \in \hat{I}} \mu\left(Q_{i}\right) \gamma_{i}} \in \Pi(\mu, \nu)$ and consequently
$$ W^{2}(\mu, \nu) \leq(c, \gamma)=\sum_{i \in \hat{I}} \mu\left(Q_{i}\right)\left(c, \gamma_{i}\right) \leq \sum_{i \in \hat{I}} \mu\left(Q_{i}\right) C^{2}=C^{2}. $$
The measure $\gamma \in \Pi(\mu, \nu)$ is defined on the whole $\mathcal X^2$, but $\operatorname{supp}(\gamma_i) \subseteq \overline{Q_{i}}^{2}$ and $\bigcup_i \overline{Q_{i}}^{2} \subsetneq \mathcal X^2$. This is why I suspect $\gamma=\sum_{i \in \hat{I}} \mu\left(Q_{i}\right) \gamma_{i}$ is not correct.
Could you please elaborate on this point?
I misunderstood the argument by the authors. Actually, we need to verify that such $\gamma$ with that particular form belongs to $\Pi(\mu, \nu)$. And for sure $W^2 (\mu, \nu)$ is not greater than the cost induced by this particular subset of $\gamma$'s.