Metric embeddings of Laakso graphs into Banach spaces

Let X be Banach space which is not super-reflexive. Then, for each n ≥ 1 and ε > 0 , we exhibit metric embeddings of the Laakso graph L n into X with distortion less than 2 + ε and into L 1 [ 0 , 1 ] with distortion 4/3. The distortion of an embedding of L 2 (respectively, the diamond graph D 2 ) into L 1 [ 0 , 1 ] is at least 9/8 (respectively, 5/4).