### Abstract

We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,..., m} ^{n}, equipped with the ℓ _{p} ^{n} metric, in any 2-uniformly convex Banach space is of order min {n ^{1/2-1/p}, m ^{1-2/p}}.

Original language | English (US) |
---|---|

Pages (from-to) | 2577-2584 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 134 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2006 |

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### Keywords

- Bi-Lipschitz embeddings
- Lipschitz extension

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*134*(9), 2577-2584. https://doi.org/10.1090/S0002-9939-06-08298-0

**Some applications of ball's extension theorem.** / Mendel, Manor; Naor, Assaf.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 134, no. 9, pp. 2577-2584. https://doi.org/10.1090/S0002-9939-06-08298-0

}

TY - JOUR

T1 - Some applications of ball's extension theorem

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2006/9

Y1 - 2006/9

N2 - We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,..., m} n, equipped with the ℓ p n metric, in any 2-uniformly convex Banach space is of order min {n 1/2-1/p, m 1-2/p}.

AB - We present two applications of Ball's extension theorem. First we observe that Ball's extension theorem, together with the recent solution of Ball's Markov type 2 problem due to Naor, Peres, Schramm and Sheffield, imply a generalization, and an alternative proof of, the Johnson-Lindenstrauss extension theorem. Second, we prove that the distortion required to embed the integer lattice {0,1,..., m} n, equipped with the ℓ p n metric, in any 2-uniformly convex Banach space is of order min {n 1/2-1/p, m 1-2/p}.

KW - Bi-Lipschitz embeddings

KW - Lipschitz extension

UR - http://www.scopus.com/inward/record.url?scp=33748354258&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748354258&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-06-08298-0

DO - 10.1090/S0002-9939-06-08298-0

M3 - Article

VL - 134

SP - 2577

EP - 2584

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 9

ER -