Ultrametric spaces bi-Lipschitz embeddable in ℝ n

Kerkko Luosto

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 1, page 25-42
  • ISSN: 0016-2736

Abstract

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It is proved that if an ultrametric space can be bi-Lipschitz embedded in ℝ n , then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in ℝ n .

How to cite

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Luosto, Kerkko. "Ultrametric spaces bi-Lipschitz embeddable in $ℝ^n$." Fundamenta Mathematicae 150.1 (1996): 25-42. <http://eudml.org/doc/212161>.

@article{Luosto1996,
abstract = {It is proved that if an ultrametric space can be bi-Lipschitz embedded in $ℝ^n$, then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in $ℝ^n$.},
author = {Luosto, Kerkko},
journal = {Fundamenta Mathematicae},
keywords = {Assouad dimension},
language = {eng},
number = {1},
pages = {25-42},
title = {Ultrametric spaces bi-Lipschitz embeddable in $ℝ^n$},
url = {http://eudml.org/doc/212161},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Luosto, Kerkko
TI - Ultrametric spaces bi-Lipschitz embeddable in $ℝ^n$
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 25
EP - 42
AB - It is proved that if an ultrametric space can be bi-Lipschitz embedded in $ℝ^n$, then its Assouad dimension is less than n. Together with a result of Luukkainen and Movahedi-Lankarani, where the converse was shown, this gives a characterization in terms of Assouad dimension of the ultrametric spaces which are bi-Lipschitz embeddable in $ℝ^n$.
LA - eng
KW - Assouad dimension
UR - http://eudml.org/doc/212161
ER -

References

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  1. [ABBW] M. Aschbacher, P. Baldi, E. B. Baum and R. M. Wilson, Embeddings of ultrametric spaces in finite dimensional structures, SIAM J. Algebraic Discrete Methods 8 (1987), 564-577. Zbl0639.51018
  2. [A] P. Assouad, Étude d’une dimension métrique liée à la possibilité de plongements dans ℝ n , C. R. Acad. Sci. Paris Sér. A 288 (1979), 731-734. Zbl0409.54020
  3. [LM-L] J. Luukkainen and H. Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994), 181-193. Zbl0807.54025
  4. [M-LW] H. Movahedi-Lankarani and R. Wells, Ultrametrics and geometric measures, Proc. Amer. Math. Soc. 123 (1995), 2579-2584. Zbl0872.54020
  5. [S] S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about A ∞ weights, Rev. Mat. Iberoamericana, to appear. Zbl0858.46017

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