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 :We prove that this is optimal up to lower order factors, i.e., sometimes max a∈N F (athen 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 n 2 → Y.