Pythagorean powers of hypercubes
Annales de l'Institut Fourier, Volume 66 (2016) no. 3, p. 1093-1116

It is shown here that for every n, any embedding into L 1 of the n-fold Pythagorean power of the n-dimensional Hamming cube incurs distortion that is at least a constant multiple of n. This is achieved through the introduction of a new bi-Lipschitz invariant of metric spaces that is inspired by a linear inequality of Kwapień and Schütt (1989). The new metric invariant is evaluated here for L 1 , implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

On montre que pour tout n, tout plongement dans L 1 de la puissance pythagoricienne n-ième du cube de Hamming de dimension n admet une distortion qui est au moins un multiple de n par une constante. Pour cela on introduit un nouvel invariant bi-Lipschitz des espaces métriques, inspiré par une inégalité linéaire de Kwapień et Schütt (1989). C’est en évaluant ce nouvel invariant sur L 1 que l’on obtient l’énoncé ci-dessus. On explique le rapport avec le programme de Ribe, et on discute des questions ouvertes.

Received : 2015-01-27
Revised : 2015-09-11
Accepted : 2015-11-25
Published online : 2016-12-14
DOI : https://doi.org/10.5802/aif.3032
Classification:  46B85,  30L05
Keywords: metric embeddings, Ribe program
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     author = {Naor, Assaf and Schechtman, Gideon},
     title = {Pythagorean powers of hypercubes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {3},
     year = {2016},
     pages = {1093-1116},
     doi = {10.5802/aif.3032},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_3_1093_0}
}
Naor, Assaf; Schechtman, Gideon. Pythagorean powers of hypercubes. Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1093-1116. doi : 10.5802/aif.3032. http://www.numdam.org/item/AIF_2016__66_3_1093_0/

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