*Accepted Paper*

**Inserted:** 17 nov 2008

**Last Updated:** 9 feb 2010

**Journal:** Rendiconti di matematica

**Year:** 2009

**Abstract:**

We prove that a $W_2$-closed, geodesically convex subset $\mathcal C$ of $P^r_2({R^d})$ is closed with respect to weak convergence in $P^r_2({R^d})$. This means that if $(\mu_n)\subset \mathcal C$ is such that $\mu_n\to\mu$ in duality with continuous bounded functions and $\sup_n \int

x

^2d\mu_n<\infty$, then $\mu\in \mathcal C$ as well.

**Keywords:**
Wasserstein distance, Geodesic convexity

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