论文标题
Brunn---inkowski不平等意味着加权歧管中的CD状况
The Brunn--Minkowski inequality implies the CD condition in weighted Riemannian manifolds
论文作者
论文摘要
由Sturm和Lott-villani开创的曲率尺寸条件CD（K，N）是一个合成的概念，它在非平滑环境中具有下面的曲率和上面的尺寸。这种情况意味着BM（K，N）表示的Brunn--Minkowski不等式的合适概括。在本文中，我们解决了在加权riemannian歧管的设置中的相反含义，证明BM（K，N）实际上与CD（K，N）相当。我们的结果允许在不使用最佳传输和歧管的差分结构的情况下表征曲率维度条件。
The curvature dimension condition CD(K,N), pioneered by Sturm and Lott--Villani, is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn--Minkowski inequality, denoted by BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of the manifold.
