Convexity of the support of the displacement interpolation: counterexamples
Résumé
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 $\mathbb W_2(\R^d)$ is necessarily convex or not. We prove that this is not the case, even when $\rho_0$ and $\rho_1$ are uniform measures.
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