Scaled Enflo Type Is Equivalent to Rademacher Type

where here, and in what follows, Eε denotes the expectation with respect to uniformly chosen ε = (ε1, . . . , εn) ∈ {−1, 1}. The infimum over all constants T for which (1.1) holds is denoted by Tp(X). An important theorem of Ribe [12] states that Banach spaces which are uniformly homeomorphic must have the same isomorphic local properties. In other words, any property which remains valid under linear isomorphisms, and whose definition involves statements about finitely many vectors, is preserved under bijections which are uniformly continuous in both directions. This fact suggests that local properties of Banach spaces have a purely metric reformulation. In the past three decades this point of view fuelled research on the metric structure of Banach spaces, and led to the development of analogs of the local theory of Banach spaces in the context of general metric spaces. We refer to the discussion in [2, 8, 9] and the references therein for more information on this topic. In particular, as explained in the above references, motivated by the search for concrete versions of Ribe’s theorem for various fundamental local properties of Banach spaces, several researchers proposed non-linear notions of type, which make sense in the setting of arbitrary metric spaces (see [5, 3, 1]). Specifically, following Enflo [5], we say that a metric space (M, dM) has Enflo type p if there exists a constant K such that for every n ∈ N and every mapping f : {−1, 1} → M, EεdM(f(ε), f(−ε))