Calculus of Variations and Geometric Measure Theory

F. Santambrogio - X. J. Wang

Convexity of the support of the displacement interpolation: counterexamples

created by santambro on 09 Oct 2015
modified on 29 Feb 2016


Accepted Paper

Inserted: 9 oct 2015
Last Updated: 29 feb 2016

Journal: Appl. Math. Letters
Year: 2016


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