### 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) |
---|---|

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 |

### Fingerprint

### Keywords

- Bi-Lipschitz embeddings
- Coarse embeddings
- Metric cotype

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

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

**Improved bounds in the metric cotype inequality for Banach spaces.** / Giladi, Ohad; Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 260, no. 1, pp. 164-194. https://doi.org/10.1016/j.jfa.2010.08.015

}

TY - JOUR

T1 - Improved bounds in the metric cotype inequality for Banach spaces

AU - Giladi, Ohad

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2011/1/1

Y1 - 2011/1/1

N2 - 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.

AB - 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.

KW - Bi-Lipschitz embeddings

KW - Coarse embeddings

KW - Metric cotype

UR - http://www.scopus.com/inward/record.url?scp=77958153940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958153940&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2010.08.015

DO - 10.1016/j.jfa.2010.08.015

M3 - Article

AN - SCOPUS:77958153940

VL - 260

SP - 164

EP - 194

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -