This is an announcement for the paper "Improved bounds in the scaled Enflo type inequality for Banach spaces" by Ohad Giladi and Assaf Naor. Abstract: 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]. Archive classification: math.FA math.MG Mathematics Subject Classification: 46B07, 46B20, 51F99 Submitted from: giladi@cims.nyu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1004.4221 or http://arXiv.org/abs/1004.4221