Coarse infinite-dimensionality of hyperspaces of finite subsets
Coarse infinite-dimensionality of hyperspaces of finite subsets
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DOI：
10.1007/s40879-021-00515-3
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发表时间：
2021-12
影响因子：
0.6
通讯作者：
Thomas Weighill;Takamitsu Yamauchi;N. Zava
中科院分区：
文献类型：
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作者：
Thomas Weighill;Takamitsu Yamauchi;N. Zava
我们考虑了由度量空间的有限子集组成的超空间在粗糙几何中的无限维性质。我们看到，通过取至多n个点的子集的超空间，几个无穷维性质得以保持。另一方面，证明了若度量空间粗嵌入长区间序列，则其有限子集超空间不可粗嵌入任何一致凸Banach空间.作为推论，真实的直线的有限子集的超空间不能粗嵌入任何一致凸Banach空间。证明了每个具有直有限分解复杂度的度量空间（不一定是有界几何）都具有度量稀疏性。
We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at mostnpoints. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.