Umbel convexity and the geometry of trees
Umbel convexity and the geometry of trees
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DOI：
10.1016/j.aim.2023.109461
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发表时间：
2021-03
影响因子：
1.7
通讯作者：
F. Baudier;C. Gartland
中科院分区：
文献类型：
--
作者：
F. Baudier;C. Gartland
对于每个p∈(0，∞)，引入了一个新的度量不变量，称为伞形p-凸性。伞形凸性的渐近概念捕捉到了可数分枝树的几何，这与马尔可夫凸性非常相似，后者是启发它的局部不变量，捕捉到有界度树的几何。Uumbel凸性被用来给出一类允许具有Rolewicz性质(β)的等价范数的Banach空间的“Poincaré型”度量刻画。我们解释了伞形p-凸性的松弛，称为下伞形p-凸性，在获得可数分枝树的粗嵌入的压缩比上界中起到了作用。引入了这些不变量的局部类似--fork p-凸性和次上fork p-凸性，并讨论了它们与马尔可夫p-凸性和p-fork不等式的松弛的关系。估计了一大类Heisenberg群的度量不变量，并证明了p-一致凸Banach空间上Heisenberg群的一个平行四边形p-凸性不等式.最后，给出了非负曲率的一个新刻画。
For every p∈(0,∞), a new metric invariant called umbel p-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a “Poincaré-type” metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property (β). We explain how a relaxation of umbel p-convexity, called infrasup-umbel p-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants-fork p-convexity and infrasup-fork p-convexity-are introduced, and their relationship to Markov p-convexity and relaxations of the p-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram p-convexity inequality is proved for Heisenberg groups over p-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.