## Appendix A Proof That the Wasserstein Barycenter Is Contracting  
Let \(\mathcal{P}^{\prime}\) be the set of discrete measures over \(\mathbb{R}^{d}\) with finite support. Let  
\[P_{X_{1}}=\sum_{i=1}^{n}\alpha_{i}\delta_{\mathbf{x}_{1}^{i}}\,\ \sum_{i=1}^{n}\alpha_{i}=1\,\ \alpha_{i}>0\]  
\[P_{X_{2}}=\sum_{j=1}^{m}\beta_{j}\delta_{\mathbf{x}_{j}^{j}}\,\ \sum_{j=1}^{m}\beta_{j}=1\,\ \beta_{j}>0\]  
\[P_{X^{*}}=\sum_{l=1}^{k}\lambda_{i}\delta_{\mathbf{x}_{l}^{*}}\,\ \sum_{l=1}^{k}\lambda_{l}=1\,\ \lambda_{l}>0\]  
Consider the Wasserstein barycenter loss function as follows:  
\[g:\left|\begin{array}{ccc}\mathcal{P}^{\prime}&\longrightarrow&\mathbb{R}^ {+}\\ P_{X}&\longmapsto&(\epsilon W_{2}^{2}(P_{X},P_{X_{1}})+(1-\epsilon)W_{2}^{2}(P _{X},P_{X_{2}}))\.\end{array}\right.\]  
The barycenter stems from the minimization of this loss,  
\[P_{X^{*}}=\arg\min_{P_{X}\in\mathcal{P}^{\prime}}(P_{X})\.\]  
We prove that the points of support of \(P_{X^{*}}\) belong to the convex closure of all the points contained in \(X_{1}\) and \(X_{2}\). The proof works by showing that a point of \(X^{*}\) cannot be outside of the closure, otherwise it yields the absurd result that it is both outside and inside. We start the proof with a complementary lemma that will soon be useful.  
Lemma A1: _If \(C\) is a convex set of \(\mathbb{R}^{d}\), so is \(\bar{C}\) (the adherence of \(C\))._  
Proof: Let \(\mathbf{x}\) and \(\mathbf{y}\) be two elements of \(\bar{C}\) and \(t\in[0,1]\). There exist two sequences \(\mathbf{x}_{n}\) and \(\mathbf{y}_{n}\) of \(C\) such that \(\lim_{n\rightarrow+\infty}\mathbf{x}_{n}=\mathbf{x}\) and \(\lim_{n\rightarrow+\infty}\mathbf{y}_{n}=\mathbf{y}\). Therefore, \(\lim_{n\rightarrow+\infty}(t\mathbf{x}_{n}+(1-t)\mathbf{y}_{n})=t\mathbf{x}+( 1-t)\mathbf{y}\). C being convex, \(\forall n\), \((t\mathbf{x}_{n}+(1-t)\mathbf{y}_{n})\in C\). We have constructed a sequence having its elements in \(C\) that tends to \(t\mathbf{x}+(1-t)\mathbf{y}\); therefore, \(t\mathbf{x}+(1-t)\mathbf{y}\in\bar{C}\).  
\[Conv(\mathbf{x}_{1}^{1},...,\mathbf{x}_{n}^{1},\mathbf{x}_{1}^{2},...,\mathbf{ x}_{m}^{2})\text{ is a convex set by definition, since it is the smallest convex set containing the points }X_{1}\text{ and }X_{2}\text{. By the above lemma, }\overline{Conv(\mathbf{x}_{1}^{1},...,\mathbf{x}_{n}^{1},\mathbf{x}_{1}^{2},...,\mathbf{x}_{m}^{2})}\text{ is convex.}\]  
Suppose that there exists a point outside of the closure of the convex set, \(\mathbf{x}_{l}^{*}\
otin\overline{Conv(\mathbf{x}_{1}^{1},...,\mathbf{x}_{n}^ {1},\mathbf{x}_{1}^{2},...,\mathbf{x}_{m}^{2})}\). \(\mathbb{R}^{d}\) equipped with the canonical scalar product (with  
By definition,  
\[\min_{\pi^{1}\in\Pi^{1},\pi^{2}\in\Gamma\Pi^{2}}h^{**}(\pi^{1},\pi^{2})=g(P_{X^{** }}),\min_{\pi^{1}\in\Pi^{1},\pi^{2}\in\Pi^{2}}h^{*}(\pi^{1},\pi^{2})=g(P_{X^{*}})\]  
\[\implies g(P_{X^{**}})\leq g(P_{X^{*}})\]  
With the hypothesis of unicity, it means that \(P_{X^{**}}=P_{X^{*}}\) and \(\mathbf{x}^{**}_{l^{\prime}}=\mathbf{x}^{*}_{l^{\prime}}\), which is a contradiction since we assume that \(\mathbf{x}^{*}_{l^{\prime}}\) is not in the closure while \(\mathbf{x}^{**}_{l^{\prime}}\) is. Therefore, \(\mathbf{x}^{*}_{l^{\prime}}\) cannot be outside of the closure, and Wasserstein barycenters produce clouds that are contracting.