强 Brunn–Minkowski 不等式及其与 CD 条件的等价性
The strong Brunn--Minkowski inequality and its equivalence with the CD condition
中文总结 AI 辅助
本文在本质非分支度量测度空间中引入强 Brunn–Minkowski 不等式 SBM(K,N)，并证明其与 Lott–Sturm–Villani 曲率维数条件 CD(K,N) 等价，为建立 CD 条件与广义 Brunn–Minkowski 不等式的完整等价性迈出第一步。
中文摘要 AI 辅助
在本质非分支度量测度空间的框架下，我们证明了 Lott–Sturm–Villani 意义下的曲率维数条件 CD(K,N) 与新引入的、我们称之为强 Brunn–Minkowski 不等式 SBM(K,N) 的概念之间的等价性。该条件是对广义 Brunn–Minkowski 不等式 BM(K,N) 的加强，而 BM(K,N) 已知在 CD(K,N) 空间中成立。我们的结果是朝着完整建立 CD(K,N) 条件与 BM(K,N) 有效性之间等价关系的第一步，这一等价关系我们最近已在加权黎曼流形框架中证明。
英文摘要
In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition CD(K,N), in the sense of Lott--Sturm--Villani, and a newly introduced notion that we call strong Brunn--Minkowski inequality SBM(K,N). This condition is a reinforcement of the generalized Brunn--Minkowski inequality BM(K,N), which is known to hold in CD(K,N) spaces. Our result is a first step towards providing a full equivalence between the CD(K,N) condition and the validity of BM(K,N), which we have recently proved in the framework of weighted Riemannian manifolds.
