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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.13
URN: urn:nbn:de:0030-drops-138129
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13812/
Backurs, Arturs ;
Mahabadi, Sepideh ;
Makarychev, Konstantin ;
Makarychev, Yury
Two-Sided Kirszbraun Theorem
Abstract
In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y.
In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖.
BibTeX - Entry
@InProceedings{backurs_et_al:LIPIcs.SoCG.2021.13,
author = {Backurs, Arturs and Mahabadi, Sepideh and Makarychev, Konstantin and Makarychev, Yury},
title = {{Two-Sided Kirszbraun Theorem}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {13:1--13:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13812},
URN = {urn:nbn:de:0030-drops-138129},
doi = {10.4230/LIPIcs.SoCG.2021.13},
annote = {Keywords: Kirszbraun theorem, Lipschitz map, Outer-extension, Two-sided extension}
}
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Keywords: |
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Kirszbraun theorem, Lipschitz map, Outer-extension, Two-sided extension |
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Collection: |
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37th International Symposium on Computational Geometry (SoCG 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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02.06.2021 |
