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 : Zn m →X we have n j=1 Ex f x + m 2 ej − f (x) q X mqEε,x f (x + ε)− f (x) q X , (1) where the expectations are with respect to uniformly chosen x ∈ Zn m 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 n 2+1 q 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 . © 2010 Elsevier Inc. All rights reserved.