This is an announcement for the paper "Improved bounds in the metric cotype inequality for Banach spaces" by Ohad Giladi, Manor Mendel, and Assaf Naor.
Abstract: 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)}.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 46B80,46B85,51F99,05C12
Remarks: 27 pages, 1 figure
The source file(s), cotypeGMN.bbl: 3212 bytes cotypeGMN.tex: 87911 bytes tr-jigsaw.eps: 52290 bytes tr-jigsaw.pdf: 28339 bytes, is(are) stored in gzipped form as 1003.0279.tar.gz with size 80kb. The corresponding postcript file has gzipped size 84kb.
Submitted from: mendelma@gmail.com
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