Pythagorean powers of hypercubes

For n ∈ N consider the n-dimensional hypercube as equal to the vector space F2 , where F2 is the field of size two. Endow F2 with the Hamming metric, i.e., with the metric induced by the `1 norm when one identifies F2 with {0, 1} ⊆ R. Denote by `2 (F2 ) the n-fold Pythagorean product of F2 , i.e., the space of all x = (x1, . . . , xn) ∈ ∏n j=1 F n 2 , equipped with the metric ∀x, y ∈ n ∏ j=1 F2 , d`2 (F2 )(x, y) def = √ ‖x1 − y1‖1 + . . . + ‖xn − yn‖1. It is shown here that the bi-Lipschitz distortion of any embedding of `2 (F2 ) into L1 is at least a constant multiple of √ n. This is achieved through the following new bi-Lipschitz invariant, which is a metric version of (a slight variant of) a linear inequality of Kwapień and Schütt (1989). Letting {ejk}j,k∈{1,...,n} denote the standard basis of the space of all n by n matrices Mn(F2), say that a metric space (X, dX) is a KS space if there exists C = C(X) > 0 such that for every n ∈ 2N, every mapping f : Mn(F2)→ X satisfies 1 n n ∑ j=1 E [ dX ( f ( x + n ∑ k=1 ejk ) , f(x) )] 6 CE [ dX ( f ( x + n ∑ j=1 ejkj ) , f(x) )] , where the expectations above are with respect to x ∈ Mn(F2) and k = (k1, . . . , kn) ∈ {1, . . . , n} chosen uniformly at random. It is shown here that L1 is a KS space (with C = 2e /(e − 1), which is best possible), implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.