### Abstract

It is shown that if (X,||·||X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m≲n^{1+1/q} such that for every f:Z_{m}^{n}→X we have. where the expectations are with respect to uniformly chosen x∈Z_{m}^{n} and ε∈{-1,0,1}n, and all the implied constants may depend only on q and the Rademacher cotype q constant of X. This improves the bound of m≲n2+1q from Mendel and Naor (2008) [13]. The proof of (1) is based on a "smoothing and approximation" procedure which simplifies the proof of the metric characterization of Rademacher cotype of Mendel and Naor (2008) [13]. We also show that any such "smoothing and approximation" approach to metric cotype inequalities must require m≳n^{1/2 + 1/q}.

Original language | English (US) |
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Pages (from-to) | 164-194 |

Number of pages | 31 |

Journal | Journal of Functional Analysis |

Volume | 260 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2011 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis

### Keywords

- Bi-Lipschitz embeddings
- Coarse embeddings
- Metric cotype

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## Cite this

*Journal of Functional Analysis*,

*260*(1), 164-194. https://doi.org/10.1016/j.jfa.2010.08.015