## Abstract

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ^{n} on the discrete cube {0, 1}^{n} and on the lattice Z^{n}. In particular we prove sharp lower estimates for the product measures of 'boundaries' of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z^{n}. The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes.

Original language | English (US) |
---|---|

Pages (from-to) | 245-277 |

Number of pages | 33 |

Journal | Probability Theory and Related Fields |

Volume | 114 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1999 |

Externally published | Yes |

## Keywords

- Concentration of measure
- Isoperimetry
- Poincaré inequalities