Transportation cost for Gaussian and other product measures

TLDR: In this paper, it was shown that the transportation cost of a unit of mass from a toy to a person, when the cost of transporting a unit is measured by ∥x−y∥2, is at most $€ 1.

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.