Pythagorean powers of hypercubes

Assaf Naor; Gideon Schechtman

doi:10.5802/aif.3032

Pythagorean powers of hypercubes

Assaf Naor, Gideon Schechtman

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

It is shown here that for every n ∈ ℕ, any embedding into L₁ 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 L₁, implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

Original language	English (US)
Pages (from-to)	1093-1116
Number of pages	24
Journal	Annales de l'Institut Fourier
Volume	66
Issue number	3
DOIs	https://doi.org/10.5802/aif.3032
State	Published - 2016

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Geometry and Topology

Keywords

Metric embeddings
Ribe program

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10.5802/aif.3032

Cite this

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AU - Schechtman, Gideon

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AB - It is shown here that for every n ∈ ℕ, any embedding into L1 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 L1, implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.

KW - Metric embeddings

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Pythagorean powers of hypercubes

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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