Discrete isoperimetric and Poincare-type inequalities

We study some discrete isoperimetric and Poincare-type inequalities for product probability measures mu(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 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 measure. A similar characterization of distributions on R which satisfy Poincare inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes.