It is shown here that for every , any embedding into of the -fold Pythagorean power of the -dimensional Hamming cube incurs distortion that is at least a constant multiple of . 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 , implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.
On montre que pour tout , tout plongement dans de la puissance pythagoricienne -ième du cube de Hamming de dimension admet une distortion qui est au moins un multiple de 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 que l’on obtient l’énoncé ci-dessus. On explique le rapport avec le programme de Ribe, et on discute des questions ouvertes.
Revised:
Accepted:
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Keywords: metric embeddings, Ribe program
Mot clés : Plongements métriques, programme de Ribe
Naor, Assaf 1; Schechtman, Gideon 2
@article{AIF_2016__66_3_1093_0, author = {Naor, Assaf and Schechtman, Gideon}, title = {Pythagorean powers of hypercubes}, journal = {Annales de l'Institut Fourier}, pages = {1093--1116}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {66}, number = {3}, year = {2016}, doi = {10.5802/aif.3032}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3032/} }
TY - JOUR AU - Naor, Assaf AU - Schechtman, Gideon TI - Pythagorean powers of hypercubes JO - Annales de l'Institut Fourier PY - 2016 SP - 1093 EP - 1116 VL - 66 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3032/ DO - 10.5802/aif.3032 LA - en ID - AIF_2016__66_3_1093_0 ER -
%0 Journal Article %A Naor, Assaf %A Schechtman, Gideon %T Pythagorean powers of hypercubes %J Annales de l'Institut Fourier %D 2016 %P 1093-1116 %V 66 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3032/ %R 10.5802/aif.3032 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.3032/
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