Hessian metrics, CD(K,N)-spaces, and optimal transportation of log-concave measures
We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M = (ℝd, g,μ). We prove, in particular, that M admits a non-negative Bakry-Emery tensor provided both V and W are convex. If the target measure ν is the Lebesgue measure on a convex set Ω and μ is log-concave we prove that M is a CD(K, N) space. Applications of these results include some global dimension-free a priori estimates of \\D2 Φ||. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for M.