Sharp Poincar\'e inequalities under Measure Contraction Property

We prove a sharp Poincar'e inequality for subsets $\mathrm{\Omega }$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\text{MCP}(K,N)$, 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 Poincar'e 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 $\mathrm{\Omega }$ is geodesically convex, by employing the geodesic hull of $\mathrm{\Omega }$ on the energy side of the Poincar'e inequality. In particular, our results apply to geodesic balls in ideal sub-Riemannian manifolds, such as the Heisenberg group.