Comptes Rendus
no. 4
Analyse mathématique, Topologie
Some comments on Laakso graphs and sets of differences
Alexandros Margaris1 ; James C. Robinson1 
1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 515-521.

We recall a variation of a construction due to Laakso [3], also used by Lang and Plaut [3] of a doubling metric space X that cannot be embedded into any Hilbert space. We give a more concrete version of this construction and motivated by the results of Olson & Robinson [6], we consider the Kuratowski embedding Φ(X) of X into L ∞ (X) and prove that Φ(X)-Φ(X) is not doubling.

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DOI : 10.5802/crmath.70
Alexandros Margaris 1 ; James C. Robinson 1

1 Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Alexandros Margaris; James C. Robinson. Some comments on Laakso graphs and sets of differences. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 515-521. doi : 10.5802/crmath.70. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.70/

[1] Patrice Assouad Plongements Lipschitziens dans ℝ n , Bull. Soc. Math. Fr., Volume 111 (1983), pp. 429-448 | DOI | MR | Zbl

[2] Juha Heinonen Geometric Embeddings of Metric Spaces, Report. University of Jyväskylä. Department of Mathematics and Statistics, 90, University of Jyväskylä, 2003 | MR | Zbl

[3] Tomi J. Laakso Plane with A ∞ -weighted metric not bilipschitz embeddable to ℝ n , Bull. Lond. Math. Soc., Volume 34 (2002) no. 6, pp. 667-676 | DOI | MR | Zbl

[4] Urs Lang; Conrad Plaut Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata, Volume 87 (2001) no. 1-3, pp. 285-307 | DOI | MR | Zbl

[5] Alexandros Margaris Dimensions, Embeddings and Iterated Function Systems, Ph. D. Thesis, University of Warwick (UK) (2019)

[6] Eric J. Olson; James C. Robinson Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Am. Math. Soc., Volume 362 (2010) no. 1, pp. 145-168 | DOI | MR | Zbl

[7] James C. Robinson Dimensions, Embeddings and Attractors, Cambridge Tracts in Mathematics, 186, Cambridge University Press, 2011 | MR | Zbl

[8] James C. Robinson Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls, Proc. Am. Math. Soc., Volume 142 (2014) no. 4, pp. 1275-1288 | DOI | MR | Zbl

[9] Stephen Semmes On the nonexistence of bilipschitz parameterizations and geometric problems about A ∞ -weights, Rev. Mat. Iberoam., Volume 12 (1996) no. 2, pp. 337-410 | DOI | Zbl

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