### 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.

Original language | English (US) |
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Pages (from-to) | 221-238 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 211 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2016 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Austin, T. (2016). On the failure of concentration for the ℓ

_{∞}-ball.*Israel Journal of Mathematics*,*211*(1), 221-238. https://doi.org/10.1007/s11856-015-1265-6