Coarse infinite-dimensionality of hyperspaces of finite subsets

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 most n points. On the other hand, we p...

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Bibliographic Details
Published in:European journal of mathematics 2022-03, Vol.8 (1), p.335-355
Main Authors: Weighill, Thomas, Yamauchi, Takamitsu, Zava, Nicolò
Format: Article
Language:English
Subjects:
Algebraic Geometry
Banach spaces
Hyperspaces
Mathematics
Mathematics and Statistics
Metric space
Research Article
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Summary: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 most n points. 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.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-021-00515-3