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 e > 0 , if N is an e-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 ≲ n e 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 ) e 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 e for some 0 c 1 2 . 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 ≲ n e . We prove that this is sharp, i.e., sometimes max a ∈ N ⁡ ‖ F ( a ) − f ( a ) ‖ Y ≳ n e for every O ( 1 ) -Lipschitz F : l 2 n → Y .