## Abstract

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small "influence sum" are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in R{double struk} ^{n} of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least ct (1 -t) √log n/n, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on R{double struk} ^{n} and the class of sets invariant under transitive permutation group of the coordinates.

Original language | English |
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Pages (from-to) | 1135-1166 |

Number of pages | 32 |

Journal | Annals of Probability |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

Externally published | Yes |

## Keywords

- Gaussian measure
- Influences
- Isoperimetric inequality
- Kahn-Kalai-Linial influence bound
- Product space