Improved bounds in the metric cotype inequality for Banach spaces

It is shown that if (X,‖⋅‖X)(X,‖⋅‖X) is a Banach space with Rademacher cotype q then for every integer n   there exists an even integer m≲n1+1q such that for every f:Zmn→X we haveequation(1)∑j=1nEx[‖f(x+m2ej)−f(x)‖Xq]≲mqEε,x[‖f(x+ε)−f(x)‖Xq], where the expectations are with respect to uniformly chosen x∈Zmn and ε∈n{−1,0,1}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≳n12+1q.