TRANSPORTATION COST FOR GAUSSIAN AND OTHER PRODUCT MEASURES

Consider the canonical Gaussian measure gamma(N) on R(N), a probabilit y measure mu on R(N), absolutely continuous with respect to gamma(N). We prove that the transportation cost of mu to gamma(N), when the cost of transporting a unit of mass from x to y is measured by parallel to x - y parallel to(2), is at most integral log d mu/d(gamma N) d mu. A s a consequence we obtain a completely elementary proof of a very shar p form of the concentration of measure phenomenon in Gauss space. We t hen prove a result of the same nature when gamma(N) is replaced by the measure of density 2(-N) exp(- Sigma(i less than or equal to N) \x(i) \). This yields a sharp form of concentration of measure in that space .