满足Sturm曲率维数条件的度量空间中有界密度的插值测度
Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
摘要：
我们在概率测度的Wasserstein空间中构造测地线，沿着该测地线，所有测度的密度都有一个由测地线端点的密度决定的上界。使用这些测地线，我们表明，一个局部Poincaré不等式和措施的收缩性质遵循的里奇曲率界定义的Sturm。我们还证明了一个大类的凸泛函，一个局部Poincaré不等式隐含的功能的弱位移凸性。
英文摘要：
We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincaré inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincaré inequality is implied by the weak displacement convexity of the functional.
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