We prove a sharp Poincare inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, 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 $\Omega$ is geodesically convex, by employing the geodesic hull of $\Omega$ 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.

中文翻译：

---

测量收缩性质下的尖锐 p-庞加莱不等式

我们证明了满足度量收缩属性 $\textrm{MCP}(K,N)$ 的（基本上是非分支的）度量空间的子集 $\Omega$ 的尖锐 Poincare 不等式，其直径在 $D$ 以上。这是通过识别相应的一维模型密度和定位参数来实现的，确保我们获得的庞加莱常数最可能作为 $K$、$N$ 和 $D$ 的函数。我们工作的另一个新特点是，通过在 Poincare 不等式的能量侧使用 $\Omega$ 的测地线外壳，我们不需要假设 $\Omega$ 是测地凸的。特别是，我们的结果适用于理想的亚黎曼流形中的测地线球，例如海森堡群。