no. 3
Pythagorean powers of hypercubes
[Puissances pythagoriciennes des hypercubes]
Naor, Assaf1 ; Schechtman, Gideon2
1 Princeton University Mathematics Department Fine Hall, Washington Road Princeton, NJ 08544-1000 (USA)
2 Weizmann Institute of Science Department of Mathematics Rehovot 76100 (Israel)
Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1093-1116.

On montre que pour tout n∈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.

It is shown here that for every n∈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.

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DOI : 10.5802/aif.3032
Classification : 46B85, 30L05
Keywords: metric embeddings, Ribe program
Mot clés : Plongements métriques, programme de Ribe

Naor, Assaf 1 ; Schechtman, Gideon 2

1 Princeton University Mathematics Department Fine Hall, Washington Road Princeton, NJ 08544-1000 (USA)
2 Weizmann Institute of Science Department of Mathematics Rehovot 76100 (Israel) 
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Naor, Assaf; Schechtman, Gideon. Pythagorean powers of hypercubes. Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 1093-1116. doi : 10.5802/aif.3032. https://aif.centre-mersenne.org/articles/10.5802/aif.3032/

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