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Local Poincaré inequalities from stable curvature conditions on metric spaces

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the Rényi entropy functional $E_N(ρm) = -\int_X ρ^{1-1/N} dm$ or the Shannon entropy functional $E_\infty(ρm) = \int_X ρ\log ρdm$. We also prove that if the Rényi entropy functional is strongly displacement convex in the Wasserstein space, then at every point of the space we have unique geodesics to almost all points of the space.