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Lipschitz clustering in normed spaces and barycentric spaces
Abstract
For every normed space X and integers 1≤ k<n, we construct a Lipschitz retraction from X(n) onto X(k) with constant at most 2+8n(n−1), independently of the dimension of X. Here X(n) consists of the nonempty subsets of X with at most n points, equipped with the Hausdorff metric. For the consecutive case k=n−1, a separate construction gives the sharper asymptotic bound O(nlog n). These results extend to barycentric metric spaces and settle some open problems on Lipschitz retractions.
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BibTeX
@misc{kovalev2026lipschitz,
title = {{Lipschitz clustering in normed spaces and barycentric spaces}},
author = {Leonid V. Kovalev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31555},
eprint = {2610.31555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31555},
}
RIS
TY - UNPB
TI - Lipschitz clustering in normed spaces and barycentric spaces
AU - Leonid V. Kovalev
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31555
UR - https://arxiv.org/abs/2610.31555
AB - For every normed space X and integers 1≤ k<n, we construct a Lipschitz retraction from X(n) onto X(k) with constant at most 2+8n(n−1), independently of the dimension of X. Here X(n) consists of the nonempty subsets of X with at most n points, equipped with the Hausdorff metric. For the consecutive case k=n−1, a separate construction gives the sharper asymptotic bound O(nlog n). These results extend to barycentric metric spaces and settle some open problems on Lipschitz retractions.
ER -