Transportation cost for Gaussian and other product measures

Abstract

Consider the canonical Gaussian measure γ _N on ℝ, a probability measure μ on ℝ^N, absolutely continuous with respect to γ _N . We prove that the transportation cost of μ to γ _N , when the cost of transporting a unit of mass fromx toy is measured by ∥x−y∥², is at most$\int {\log \frac{{d\mu }}{{d_{\gamma N} }}d\mu }$ dμ. As a consequence we obtain a completely elementary proof of a very sharp form of the concentration of measure phenomenon in Gauss space. We then prove a result of the same nature when γ _N is replaced by the measure of density 2^−N exp (− ∑ _i≤N |x _i |). This yields a sharp form of concentration of measure in that space.