Impossibility of almost extension

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^cε for some [Formula presented]. If X=ℓ₂ⁿ, 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:ℓ₂ⁿ→Y.