Discrete isoperimetric and Poincare-type inequalities

We study some discrete isoperimetric and Poincare-type inequalities for pro duct probability measures mu(n) on the discrete cube {0, 1}(n) and on the l attice Z(n). In particular we prove sharp lower estimates for the product m easures of 'boundaries' of arbitrary sets in the discrete cube. More genera lly, we characterize those probability distributions mu on Z which satisfy these inequalities on Zn. The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential mea sure. A similar characterization of distributions on R which satisfy Poinca re inequalities on the class of convex functions is proved in terms of vari ances of suprema of linear processes.