Abstract
Let X be Banach space which is not super-reflexive. Then, for each \(n\ge 1\) and \(\varepsilon >0\), we exhibit metric embeddings of the Laakso graph \({\mathcal {L}}_n\) into X with distortion less than \(2+\varepsilon \) and into \(L_1[0,1]\) with distortion 4/3. The distortion of an embedding of \({\mathcal {L}}_2\) (respectively, the diamond graph \(D_2\)) into \(L_1[0,1]\) is at least 9/8 (respectively, 5/4).
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Acknowledgements
We thank Mikhail Ostrovskii for his comments and for drawing some references to our attention. S. J. Dilworth was supported by Simons Foundation Collaboration Grant no. 849142. Denka Kutzarova was supported by Simons Foundation Collaboration Grant no. 636954.
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Communicated by Mikhail Ostrovskii.
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Dilworth, S.J., Kutzarova, D. & Stankov, S. Metric embeddings of Laakso graphs into Banach spaces. Banach J. Math. Anal. 16, 60 (2022). https://doi.org/10.1007/s43037-022-00212-7
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DOI: https://doi.org/10.1007/s43037-022-00212-7