On the failure of concentration for the ℓ_∞-ball

2013/09/12 by Tim Austin, Austin, Tim
Mathematics · #25D07 (primary) #47D07 #51F99 (secondary) #60E15 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:25D07 #msc:47D07 #msc:51F99 #msc:60E15

paper · pdf · doi:10.48550/arxiv.1309.3315

13 pages; [Jun 23rd, 2014:] Section 2 strengthened following referee's suggestions

arxiv created 2014/06/23 · arxiv updated 2014/06/24

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 X^N of copies of d, and consider 1-Lipschitz functions X^N→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 X^N. 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.

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