Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem

Abstract

We prove that the suitably defined surface area of a subsetA of the cube {0,1}n is bounded below by a certain explicit function of the size ofA. We establish a family of logarithmic Sobolev inequalities on the cube related to this isoperimetric result. We also give a quantitative version of Margulis' graph connectivity theorem.