  It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge
1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C
is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+
m\e/2)-f(x)}||_X^p] < C(p,X) m^p\sum_{j=1}^n\Avg_x[||f(x+e_j)-f(x)||_X^p],
where the expectation is with respect to uniformly chosen x \in Z_m^n and \e
\in \{-1,1\}^n, and C(p,X) is a constant that depends on p and the Rademacher
type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained
in [Mendel, Naor 2007]. The proof is based on an augmentation of the "smoothing
and approximation" scheme, which was implicit in [Mendel, Naor 2007].
