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So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$

These have symmetry (exchanging $\mu_1^j$ by $\mu_2^j$) such that we can say that they're invariant under the action of $\mathbb{Z}/2\mathbb{Z}$.

If now, I define $\bar{\rho} = \operatorname{argmin}_{\rho} \sum W_2^2(\rho_j,\rho) $ where $W_2$ is the two-Wassertein distance. I would like to prove that $\bar{\rho}$ has the same invariance as the $\rho_j$'s.

In order to do so, I tried to say that this is a strictly convex problem so that the minimizer is unique. Then I tried to prove that the sums are equals by proving $$W_2^2(\bar{\rho}(g \cdot), \rho_j)=W_2^2(\bar{\rho}, \rho_j(g \cdot) $$ Where $g$ is an element of the group. I don't know how to prove this ... Moreover, I think that we can prove $$\exists (\mu_1,\mu_2):\bar{\rho}=1/2\delta_{\mu_1}+1/2\delta_{\mu_2} $$

But I couldn't find any proof. Does anyone have some hints, help or thoughts about these two things ?

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