## 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 |
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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 - 1 Jan 2011 |

### Bibliographical note

Funding Information:O.G. was partially supported by NSF grant CCF-0635078. M.M. was partially supported by ISF grant no. 221/07, BSF grant no. 2006009, and a gift from Cisco research center. A.N. was supported in part by NSF grants CCF-0635078 and CCF-0832795, BSF grant 2006009, and the Packard Foundation.

## Keywords

- Bi-Lipschitz embeddings
- Coarse embeddings
- Metric cotype