## Abstract

Let (X,‖⋅‖_{X}),(Y,‖⋅‖_{Y}) be normed spaces with dim(X)=n. Bourgain's almost extension theorem asserts that for any ε>0, if N is an ε-net of the unit sphere of X and f:N→Y is 1-Lipschitz, then there exists an O(1)-Lipschitz F:X→Y such that ‖F(a)−f(a)‖_{Y}≲nε for all a∈N. We prove that this is optimal up to lower order factors, i.e., sometimes max_{a∈N}‖F(a)−f(a)‖_{Y}≳n^{1−o(1)}ε for every O(1)-Lipschitz F:X→Y. This improves Bourgain's lower bound of max_{a∈N}‖F(a)−f(a)‖_{Y}≳n^{c}ε for some [Formula presented]. If X=ℓ_{2}^{n}, then the approximation in the almost extension theorem can be improved to max_{a∈N}‖F(a)−f(a)‖_{Y}≲nε. We prove that this is sharp, i.e., sometimes max_{a∈N}‖F(a)−f(a)‖_{Y}≳nε for every O(1)-Lipschitz F:ℓ_{2}^{n}→Y.

Original language | English (US) |
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Article number | 107761 |

Journal | Advances in Mathematics |

Volume | 384 |

DOIs | |

State | Published - Jun 25 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- Lipschitz extension