### 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) |
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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 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Bi-Lipschitz embeddings
- Lipschitz extension

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## Cite this

Mendel, M., & Naor, A. (2006). Some applications of ball's extension theorem.

*Proceedings of the American Mathematical Society*,*134*(9), 2577-2584. https://doi.org/10.1090/S0002-9939-06-08298-0