Improved bounds in the metric cotype inequality for Banach spaces

It is shown that if (X,||·||X) is a Banach space with Rademacher cotype q then for every integer n there exists an even integer m≲n^1+1/q such that for every f:Z_mⁿ→X we have. where the expectations are with respect to uniformly chosen x∈Z_mⁿ and ε∈{-1,0,1}n, and all the implied constants may depend only on q and the Rademacher cotype q constant of X. This improves the bound of m≲n2+1q from Mendel and Naor (2008) [13]. The proof of (1) is based on a "smoothing and approximation" procedure which simplifies the proof of the metric characterization of Rademacher cotype of Mendel and Naor (2008) [13]. We also show that any such "smoothing and approximation" approach to metric cotype inequalities must require m≳n^{1/2 + 1/q}.