On the failure of concentration for the ℓ -ball

Tim Austin

Research output: Contribution to journalArticle

Abstract

Let (X, d) be a compact metric space and µ a Borel probability on X. For each N ≥ 1 let d N be the ℓ-product on XN of copies of d, and consider 1-Lipschitz functions XN → ℝ for d N. If the support of µ is connected and locally connected, then all such functions are close in probability to juntas: that is, functions that depend on only a few coordinates of XN. This describes the failure of measure concentration for these product spaces, and can be seen as a Lipschitz-function counterpart of the celebrated result of Friedgut that Boolean functions with small influences are close to juntas.

Original languageEnglish (US)
Pages (from-to)221-238
Number of pages18
JournalIsrael Journal of Mathematics
Volume211
Issue number1
DOIs
StatePublished - Feb 1 2016

ASJC Scopus subject areas

  • Mathematics(all)

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