Lipschitz Clustering in Metric Spaces

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipsc...

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Bibliographic Details
Published in:The Journal of geometric analysis 2022-07, Vol.32 (7), Article 188
Main Author: Kovalev, Leonid V.
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Clustering
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Mathematics
Mathematics and Statistics
Metric space
Obstructions
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Summary:In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-022-00928-w