Impossibility of almost extension

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 maxa∈N⁡‖F(a)−f(a)‖Y≳n1−o(1)ε for everyO(1)-Lipschitz F:X→Y. This improves Bourgain's lower bound of maxa∈N⁡‖F(a)−f(a)‖Y≳ncε for some 0<c<12. 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 everyO(1)-Lipschitz F:ℓ2n→Y.