Improved bounds in the scaled Enflo type inequality for Banach spaces

2010/04/23 by Ohad Giladi, Giladi, Ohad, Assaf Naor +1
Mathematics · #46B07 #46B20 #51F99 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG #msc:46B07 #msc:46B20 #msc:51F99

paper · pdf · doi:10.48550/arxiv.1004.4221

arxiv created 2010/04/23 · arxiv updated 2010/04/27

Abstract

It is shown that if (X,||.||_X) is a Banach space with Rademacher type p ≥ 1, then for every integer n there exists an even integer m < Cn^2-1/plog n (C is an absolute constant), such that for every f:Z_mⁿ --> X, \Avg_x,\e[||f(x+ m\e/2)-f(x)||_X^p] < C(p,X) m^p∑_j=1ⁿ\Avg_x[||f(x+e_j)-f(x)||_X^p], where the expectation is with respect to uniformly chosen x ∈ Z_mⁿ and \e ∈ \-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].

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