# Local Poincaré inequalities from stable curvature conditions on metric spaces

**published**
**year:** 2012

**journal:** Calc. Var. Partial Differential Equations

**abstract:** 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 $\mathcal{E}_N(\rho m) = -\int_X \rho^{1-1/N} dm$ or the Shannon
entropy functional $\mathcal{E}_\infty(\rho m) = \int_X \rho \log \rho dm$.
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.

The paper is available on the
cvgmt preprint server.