Convexity of the support of the displacement interpolation: Counterexamples

Given two smooth and positive densities ρ0,ρ1ρ0,ρ1 on two compact convex sets K0,K1K0,K1, respectively, we consider the question whether the support of the measure ρtρt obtained as the geodesic interpolant of ρ0ρ0 and ρ1ρ1 in the Wasserstein space W2(Rd)W2(Rd) is necessarily convex or not. We prove that this is not the case, even when ρ0ρ0 and ρ1ρ1 are uniform measures.