Improved bounds in the metric cotype inequality for Banach spaces

Ohad Giladi, Manor Mendel, Assaf Naor

Research output: Contribution to journalArticle

4 Scopus citations

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≲n1+1/q such that for every f:Zmn→X we have. where the expectations are with respect to uniformly chosen x∈Zmn and ε∈{-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≲n2+1q from Mendel and Naor (2008) [13]. The proof of (1) is based on a "smoothing and approximation" procedure which simplifies the proof of the metric characterization of Rademacher cotype of Mendel and Naor (2008) [13]. We also show that any such "smoothing and approximation" approach to metric cotype inequalities must require m≳n1/2 + 1/q.

Original languageEnglish (US)
Pages (from-to)164-194
Number of pages31
JournalJournal of Functional Analysis
Volume260
Issue number1
DOIs
StatePublished - Jan 1 2011
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis

Keywords

  • Bi-Lipschitz embeddings
  • Coarse embeddings
  • Metric cotype

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