*Accepted Paper*

**Inserted:** 9 oct 2015

**Last Updated:** 29 feb 2016

**Journal:** Appl. Math. Letters

**Year:** 2016

**Abstract:**

Given two smooth and positive densities $\rho_0,\rho_1$ on two compact convex sets $K_0,K_1$, respectively, we consider the question whether the support of the measure $\rho_t$ obtained as the geodesic interpolant of $\rho_0$ and $\rho_1$ in the Wasserstein space $W_2(\mathbb R^d)$ is necessarily convex or not. We prove that this is not the case, even when $\rho_0$ and $\rho_1$ are uniform measures.

**Keywords:**
displacement convexity, convex bodies

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