TY - JOUR

T1 - Impossibility of almost extension

AU - Naor, Assaf

N1 - Funding Information:
Supported in part by the BSF , the Packard Foundation and the Simons Foundation . The research that is presented here was conducted under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank.
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/6/25

Y1 - 2021/6/25

N2 - 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 maxa∈N‖F(a)−f(a)‖Y≳n1−o(1)ε for every O(1)-Lipschitz F:X→Y. This improves Bourgain's lower bound of maxa∈N‖F(a)−f(a)‖Y≳ncε for some [Formula presented]. If X=ℓ2n, then the approximation in the almost extension theorem can be improved to maxa∈N‖F(a)−f(a)‖Y≲nε. We prove that this is sharp, i.e., sometimes maxa∈N‖F(a)−f(a)‖Y≳nε for every O(1)-Lipschitz F:ℓ2n→Y.

AB - 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 maxa∈N‖F(a)−f(a)‖Y≳n1−o(1)ε for every O(1)-Lipschitz F:X→Y. This improves Bourgain's lower bound of maxa∈N‖F(a)−f(a)‖Y≳ncε for some [Formula presented]. If X=ℓ2n, then the approximation in the almost extension theorem can be improved to maxa∈N‖F(a)−f(a)‖Y≲nε. We prove that this is sharp, i.e., sometimes maxa∈N‖F(a)−f(a)‖Y≳nε for every O(1)-Lipschitz F:ℓ2n→Y.

KW - Lipschitz extension

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U2 - 10.1016/j.aim.2021.107761

DO - 10.1016/j.aim.2021.107761

M3 - Article

AN - SCOPUS:85104484419

VL - 384

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107761

ER -