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We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentiallynon-branching) metric measure spaces satisfying the Measure ContractionProperty $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This isachieved by identifying the corresponding one-dimensional model densities and alocalization argument, ensuring that the Poincar\'e constant we obtain is bestpossible as a function of $K$, $N$ and $D$. Another new feature of our work isthat we do not need to assume that $\Omega$ is geodesically convex, byemploying the geodesic hull of $\Omega$ on the energy side of the Poincar\'einequality. In particular, our results apply to geodesic balls in idealsub-Riemannian manifolds, such as the Heisenberg group.

Sharp Poincar\'e inequalities under Measure Contraction Property