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}$such that for every f:Z_m^n --> X we have$\sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x) ||_X^q ] < C m^q \Avg_{\e,x} [ ||f(x+\e)-f(x) ||_X^q ]$, where the expectations are with respect to uniformly chosen x\in Z_m^n and \e\in \{-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+\frac{1}{q}} from [Mendel, Naor 2008]. The proof of the above inequality is based on a "smoothing and approximation" procedure which simplifies the proof of the metric characterization of Rademacher cotype of [Mendel, Naor 2008]. We also show that any such "smoothing and approximation" approach to metric cotype inequalities must require m> n^{(1/2)+(1/q)}.