Sharp Poincare inequalities under Measure Contraction Property

We prove a sharp Poincare inequality for subsets ω of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property MCP(K, N), whose diameter is bounded above by D. This is achieved by identifying the corresponding one-dimensional model densities and a localization argument, ensuring that the Poincare constant we obtain is best possible as a function of K, N and D. Another new feature of our work is that we do not need to assume that ω is geodesically convex, by employing the geodesic hull of ω on the energy side of the Poincare inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.