*Published Paper*

**Inserted:** 29 nov 2012

**Last Updated:** 25 aug 2014

**Journal:** Bull. London Math. Soc.

**Year:** 2013

**Abstract:**

Let $(X,d,m)$ be a geodesic metric measure space. Consider a geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space. Then as $s$ goes to $t$, the support of $\mu_{s}$ and the support of $\mu_{t}$ have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of $(X,d,m)$ and $t\in[0,1]$, we consider the set of times for which this geodesic belongs to the support of $\mu_{t}$. We prove that $t$ is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying $\mathsf{CD}(K,\infty)$. The non branching property is not needed.

**Keywords:**
optimal transportation

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